T6 rrtrrd +1 m vngt t, si"r 61 rrTq iteTr s€t (v) sinze +cos20 = 1; sin2e =1-cos2Q icos20 = 1-sinze

sqf*6 vri er w t+,q qnt (vi) sec20 = 1 + tan2Q i sec2g - tanze = 1; tanze = sec20 -
ffiurtft t rf'dflsrd reaqut q* 1
(vii)coseczg = 1 +cot2Qi cosec2Q -cot2e=1; cot26
1. 1gr+tq = 9oo
1o = 60' = cosec2 e -1
1'= 60"
(viiiltan, = lcoto =
1qq+tq = 100s # #
18=
1'= 100"
100' ur. ffiqFrfr rf.t'sfr rmFr * fqa ,
10\ \ 9G reerfu I II Iil IV
3. D"=tTg,GTg = fen
'o Sin, Cosec + +
4. 18oo = , tlg+t + +
cos, sec
1 tBfi = b7" !7'4b" qI 2o626b ++rg
tan, cot + +

5. 1" =
fa

d- = 0.01745 tfsm (etrw; rv. ffiurfrrfr wnefr sr{qrd * qrt,

30q gl 45q ql 60e ql 900 ql 1204 q 1354 q 1500 q 1804 ql
rD +iur
6. Do=
180
tfs+t 0o
T
6
fi
4'
1l

T
fi
,
2n
?
3n
i
5n
A
1t

1 I 6 J5 1 1

s 22
(etW;
Sin 0
, Jz 2.
1
2 6 , 0
t. fi-
T = 3.1416 E
v3 1 1
_, 1 1
_Jt
cos 1
T Jz , 0
J2 )
-1
1 = 0.31831 (ErrrW;
fi
tan 0 E
1
1
.6 ,tYt(Hil q(l -f -1 -6 1
0

I 1 I

8. e (tfsfi {) =! 6tu:# qr cot J5

6 0
Jr -1 -6 qt(nilqq

sec I
2
Ja J' 2 srqftqrFqr -2 _J' -6 2
-1
(n-2)
qn -
9. €T{gTq
qd = Tqtei 61 Hqr
d rdo er<: qtq qr x18oo cosec eTqfo{rfu 2 J' 6
2
I
r 2
J' 2 3TqRqrfrd

"
10. T{ E1 qRlq (qf( tf,qr
v. tdF{:{ q}ur * ffiurfift-F.iefr o{rnir:
"d) = 2nr (-o) (90"-o) (90"+O) (r8r-o) (1t0"+0) (360'-0) (36d+0)
p bp
II. (il sinO - h i cosQ - ; itane= b sln -eine cose cos0 sinO - eino - sin0 sinO

hhb cos coeO sinO -sine -cose -cosO cosO cos0

cosec Q =-iseCe=:-iCOtfl=-
p"b-p tan - tan0 cot0 -cot0 -tanO tanO -tane tanO

cosec - coseco secO secO cosec0 -coseco -cosece cosecO

sec sec0 cosecO - cosec0 - sec0 -secO secO sec0

cot -cot0 tane -tan0 -cot0 cot0 -coto cot0

y1. {C5' qf qtE IIffi ?f,tuf (Complementary angle) :

b
qf( d 6M sl +q eo'qr sr+tur d, d Bt EtE {r+, 6tur

(ii) sinQ 1 qri tr

. coseco = 1i sino i cosec6 =
ri"e *+- e HsTr (e0. - e) qfi {gt + +tE 1r+ dq ffiqrtil
(iii)coso. seco = 1; cosg - #isecQ = # VII.HuIEI' *uf (Suplementary angle) :

qfq d ffi 6r +q 180'd d Y0 vqro oiq q,rd t'r

(iv) tan0. boto = 1; tano =
# i cotg =# +A- e *{ (r80'- o) qs-{€i + Hqr+. otur *.aertir

--1
,,i ,, ::'::: - S.S.C. ADVANCED MATHEMATICS
 qd e +ttTc+!,8*rrn*1
J5-r 2'2
1. sinlbo = -rE = cos75o o+{

A+Brrn*r ffisefuo+fdq
t'6 +r
2. cosl5o = 2E = sin75o
2

tanA+tanB+tanC - tanA tanB tanC

3. tan15o=2-Jl = cos75o 7. tan(A+B+C)= 1 - tanA.tanB-tanB. tanC - tanC.tanA
4. cotlSo=2+J, = tan75o 8. sin (A+B). sin(A -B) = sin2A - sinzB = cos2B - eos2A
.6-r 9. cos (A + B) . cos (A- B) = cos2A - sin2B = cos2B - sin2A
5. sinlSo = = cos72" x. {unsa ql qfiEdq *rr qr g,t<R fr :

1. 2sinA cosB = sin (A + B) + sin (A - B) tA > Bl

Jro + zJs 2. 2 cosAsinB = sin (A+ B) - sin (A- B) [A> B]
6. coslSo = = sin72o 3. 2 cosA cosB = cos (A + B) + cos (A _ B)
- 4. 2sinA sinB = cos (A - B) - cos(A + B)

7. sinzz
!" = b. sinC + sinD = 2 sin
?. .o. f
CID
8. cos22;" = 6. sinC - sinD = 2"o"ff. .i.

9. tan22;' =,,12-t 7. cosC + cos f = 2cos +. "o, f

10. cot 22;" = Ji * t 8. cosc - +.
cosD = 2sin .i" T
xI. ffiurfirfr rf'd* *ur * cqurs { Igrq ,

11. sin36o = llr-{5 = cos54" 1. 2e'* T- srtqrd -

4
2tan9
(a) sin20 =2sin0.cosg =
.,6 *r 1+tan20
L2. cos36o = = sin54o (b) cos2 0 = .o.2 e - sin20 = 1- 2 sin2g
1-tan20
13. sin9o = Js*Js -Js-Js = cos 81o =2cos20 - ]. =
- 4 - 1.tr;%
..........................:

Js+Js +Js-Js (c) tan2o = qd e s (2n +D

14. cosgo - 4
= sin 81o #* ;
(d) 2sin2o =1-cos2o
15. cos 36o - cos72'= L (e) 2cos20 =1*cos20
z. Be + T- or1qril-
cos 36o . cosl2o = !
(a) sin30 =3sine -4sin3e
16.
4 (b) cos30 = 4cos30 - 3cos0
IX. ffiqftfr-w4frfrf*d*,tq '
1. sin (A + B) = sinA cosB + cosA. sinB (c) tan3o - 3tano-ta^3O
2. sin (A - B) = sinA cosB - cosA . sinB 1- 3 tan20
3. cos (A + B) = cosA cosB - sinA sinB xrr.ffiurffi Tfdefr q}ur eTr{do.* ,
4. cos (A - B) = cosA cosB + sinA sinB
tanA+tanB 2tan9
5. tan(A+ B) =l_t.."AJr"B 11 2
(a) sin 0 =2sin2 0 cosre - 1+tan29
2
tanA-tanB
6. tan(A-B)= 1+tanA.tanB
(b) cose=cosz; -sinzf,=r-zri,,f,

--2--
ED MA
 1- tan2 9
ix) a sin0+ b cose _,tP;P J?-T
.e
=2 cos" , -l= 1+tan29
2
d b sin6+ a cosQ _^IP;F JilP
2 xil m sine+ n sine _11a1+n) (m+n)
xiil mcosg+nsinQ -(m + n) (m+n)
ztan9
(c) tano =
2 xiii)asino-bcosQ -,tre 't^';P
1_ t^nzf, xiv) b sin Q -a cos Q
_\tP;P ,tJ;P
xv) a sin2g+b coszQ 2 Jab f{*-rdr rfr'qr rFFdT
.e0
cot':-1 xvi) acoszO + bsec2 O 2 Jab t{frrdr rfr q rrh'dr
2
(d) coto - zcot9 xvii) atanr g +bcot2 6 2 Jab f{frrdr rfr q rFrirl
2
xviii) sin Q .cos Q - tlz u2
(e) 1-cosg = 2sin2
e
xix) sin g * cos Q _J'
,
xx) asin2 g + bcos2 0 b :
e
(f) 1tcoso =2cos2 a>b
,
XIII. xxi) asinz 6 + bcosz Q b

ab acb
(a) ,ln" = #. = # = 2R drlT, wi' n Tf, 61 t*qr tr
SOME IMPORTANT TRICKS
b2 + c2 - a2 +
(b) (i) cosA = (ii) cos B "2 ^2 -b2 1. msinQ + ncosQtFI rffifq qH = Jmz+n2
2bc N^
2. tanlo.cos2o .............. tan89o=1
(iii) cosC =
a2 +b2 -c2 3. cotlo. cot2" cot89o = 1
2ab 4. coslo. cos2o cos9Oo = 0

cosin"W rft HftIrdT ttit

lz16 iiOt 5. coslo. cos2o........ (cos 90" lt e{iffi) = O
6. sinlo. sin2o sin180o = 0
xrv. ffiqfirfrq 3r{trrfr a,r qRF{ (Range) :
sinlo. sin2o
(il i5l qH 6+W +1 sil{ -1* fiq *6 tr 7. to (sin180" t efftffi) = 0
sinQdetrT cosQ
elrf-(-1 <sing SliTeTI -1(cose <1 8. sinQ.sin20.sin4Q =7
1
sinS0
(ii) cosec e deTI sec Q ?5T rIf;T -1 t 3[ef nedf +1t q-g-f ER
-t d v+ar tr 9. cosQ. cos2S. cos4Q
1
= 7cos30
slqt-( coses e > 1 iletrI coses 0 < -1 (q .".e > l iTeiTI 10. tan0 .tan20. tan40 = tan3 0
sec0 <-1
(iii)tane dql cotgthT ITFI - -t + ofth'E6 rt d VOen SOLYED
tr
xv. ffiurffiq wtr+ $ q6rrl iln qrf,q Et=r (Maximum 1. qR stanQ =+,Ei d **='q=# 6T qFT Fr6rd-
and Minimum value of Trigonometric Functions).
Trigonometric Minimum Maximum Eet:3tanO= =tanQ ;
Function
il sinO W,
-1 1

iil sink g -1 1 3sin0 + 2cos0

iii) ksin 0 -k k 3sin0 - 2cos0
iv) Cos e 3tan0 + 2
v) Cosk e
-1 1
3tan0 -2
t e[qr dsTt 6t 61 .or e t qFr ti qrl
-1 1

vil kcos e -k k
viil ksin n e -k k
6
viiil kcos n O -k k
,-o
[ '.' tan g =f,t

--3--
S.S.C, ADVANCED MATHEMATICS
2. qR zsinze+3cos20=4,d trr,et5T qH Fft5-fd- o =30o
EFI: 7sinze+3coszg-4 1

4sin2e + 3 sin2e + 3cos2 0= 4

cosB = ;
4sin26 + 3(sin2g + cos20 ) = 4 cosP = cos60"
+ 4sinzg + 3 = 4 [....sin29 + cos20 =1] F =60o
.o 1 ( a+ B)=30'+60o=90o
srn'0 -7 6. fr (rirrQ -coso)=o,d (.ir,ao *cos40)=?
coszg-1-sinze Efr :si
13
_l

-l- -44
. sin2o L ..4
tanzg =;Ze- = Z'5 = (sin45')a + (cos4S')a

-1 - (r)a /r\a
tan"O
5 - lJr) - [€./
-
l-l r l-l

1 111
tang = 15' - 4' 4- 2

frtane . # =2 d, d tanzo .#e 6l qH Fffird- l.

E - sine
.i:.il
\l1+sin0
q-{ fiffird

EFt: tano * .-L^

tanO -, EFI:
+ tan2g*1=2tan0
Ji-silr0 fi-sine
+ tan26 -2tanQ +1=0 - Ji +.*e Ji - sme
(tan g -1)2 = 0
= 1-sinO 1-sino
= tanQ -1 =0 = ;------;_
sin"e = Vcos"o
r/1 -
,-_--F-
tan0- 1
1- sin0 1 sin0
tan2g.#=r*| -2 cos0 - cosO-cosO
= secO - tan0
1
4. cos23oo. cos24bo + 4sec26o" + iz cos2go" - 2tan26o'H'[
Fr+.rd-

EFI : cos2goo. cos24bo+4sec260" *f,cos2g0o -2tanzilO" EFI :

.Eoll +-
.,6;0+1
= 22)+(;"')-z' (JE)' - J*.e+1
[+)' [#)'+(4x Jsece-1

16+0-6 =@-r)+(secO +r)

(secO

83 2sec0 2sec0
r-------i- l
& =
ri/sec'0-l r/tan"0
1-.
=rfi cos B
1
=7 d, d ( o + B) = ?
2secO
= t.rre = 2sec 0 . cot 0
2 cosO 2
= .o.o " .irrg = ,irre = 2cosec o
sin30o 9. fr.cosS -bsinO =",d asine +bcos0 -?
6Ft : (a cos0 -bsin0)2 + (a sine +b coso)2 14. qR sin BA=cos (A- 10'), qd saqfi IFo 6iq d, d a
= a2 cos2 e + b2 sin2 g + a2 sin2 g + bzcos2 Q ?5't TIFT fiffiTd_
= a2 (cos2g + sin20 ) + bz (cos2g + sin2e ) Eft : sinSA = cos (A - rO')
= cos (90'- = cos (A - gA) 109
- a2 +b2 ['.' cos2g + sin20 = 1]
(a
+ "2 + sine + b cose )2 = a2 +b2 = 90o-3A=A-10o
+ 4A= 100'
['.' (a coso - bsino )- c] ... A= 25"
(asinQ +bcoso)2= a2+b2-c2 15. cotl2o. cotSSo cot 52" cot60" cot7So =
= ?

...asine +bcosS = +J;\b2<2 6Ft : cot12o.cot38" cot52" cot60o cot7So

= (cot12"cot78"). (cot38'. cot52o) . cot60o
10. qR cose +sinQ -.f cose,d cosQ -sing -? = [ cotl2o cot (90" - 129]. [cot38' . cot(90" - 38')] cot60o

EFt : cose +sinQ -.fcosQ = [cot12o . tanl2o] [cot38". tan38'] cot60"

1
+ (cosg + sin0 )2 = (JZ cose )2 =1x1"J5 [... cot. tan = 1 & cot6o" =] I

= cos2g + sin2e + 2cos0. sino = 2cosze

1
=cos20 - sin2 0 = 2sin0 . cosg
JE
+ (cosg + sinQ ) (cos0 - sing ) = 2sin0 .cosg
16. qR cosgcr =sino *t go <90'd, d tan5cr =?
- (J-Z cos0 ) (cosg - sin 0) = 2 sin0 . cosg EeI: cos9cr =sino
2sinO.cos0 + cosgo = cos (go" - o )
=(cosO -sino)- .fzcose
= 9o =90o- o
.'. (cosO
+ 10o = 90"
-sine)= ..[ sin g cr =9o
11. qR sin Q + sin2 0 = 1d, d cosze + cos4e t5I rtFT
tan 5 cr
fir+rd- = tan45o
EFI :+ sing + sin20 = 1 =l
+ sinQ -1-sinzo t7 ffi.o.(cr + p)=0 d, d sin( o- p)=?
+ sinQ - cos2e 6Gt: cos(cr+p)=0
:+ cos( cr+ p)=cos9Oo
= sin2 e = tot4 e
+ (1-cos20)=cosa0 =) o+ I =90"
cosze +cos40 =1 + o =(go"- p)
12. qR x=rsino cosp,y=rsino sinB et{ z=rcosod, d ( o - B)=(90'-2P)
**y'+22=? = sin(cr - F)=sin(90'-2p)
Edt : x2 + yz + z2 =r2 sin2cr cos2 g+ *2sin2o sinzB +12cos2cr, sin (cr - F) = cos2 B
zz = r2 sin2a
+ yz + (cos2B+ sin2p) + r2 cos2 o
= x2 TE 1I TE

+ yz + zz = r2 sin2 a* 12 cos2ct, 18. tan

; tan ,, tan g =?
=x2
+ y2 + zz = r2 (sin261 * coszcr )
= x2 EFI : tan 5o tan 10o tan 20o
.'.*2+y2+22=t2
qR =tanBxS(qdg =S)
13. sinQ +cosQ-.f cosed, d cotQHI qH fTtF,fA- = tan 15o

EGt : sinQ +cosQ = .f cosg =2_ JB

sin0+cosO 5cos0 Tc 5n 7n
= .i"e =\2.*e 19. sin ,, sin ,* sin ,, =?
= 1*cotQ =.,[cotg ltr 180ox5 180'x7
EFt : sin O sin ,, sin ,,
= (Jr-1)coto=1 = sinlOo . sin5Oo . sin70'
, (f +r) = sin (90'- 80') sin(90'- 40"). sin (90" - 20)
+cote=ml"b; = cos80o. cos40o. cos2Oo
= cos2Oo . cos4Oo . cos80"
.'. cote = (.E+r)

--5--
S.S.C. ADVANCED MATHEMATICS
 1 1* tan7So'
- zcos60o
23. ffi $I rTFT Srd Ht-
111 1+ tan75o
42 8
EFT : 1- tan?b'
zo. AB=4ffi., BC=8tfr. *t ac=5ffi. d, d tan45o rtanTSo
^ABCt,
tFT rTH EFIT frnZ
cos B 1 - tan45".tan75o
_b2 = tan(45o+75")
a2 + c2
= tan120"
= tan (t80"
- 609
_ 82 +42 -sz =_tan60"=_.8
2x8x 4 24. qR r *tan2x-2tanx= g e1tt 0 < x < g0od, d ru5t
64+t6-25 55 Hr drn?
6eT: 1*tan2x-2tanx=O
+ (t - tanx)2 = o
21' qR sinO* 1 r cosO =4 EI' dI O t5,T TTFT F]6Td' qEi,
= (t-tarrx)=o
1+ .irrg
"r"o + tan-x = 1
+ tan-r = tan45o [ '.' 0 <, 90" l
.'. x = 45"

sinO 1* cosO 1.\^1

EFI: 1+.".e-.irre =4 25. qR *+1xx-=2cosed, d ,3+ itrl rIFT SIil q,i-

1
sin2e+(r+coso)2
- , EFI: x+-x =2cosQ
' (t + cos0Xsin0)

sin2e+1+cos2o+2coso
-4 + /[,.;J
r)3
= (2cose )B
(t + cos0)(sinO)

+
sin20+cos20+1+2cos0
(t + cos0)(sin0)
-4 = ,r*i.r[*i) =scos'o

1+1+2cos0 i 3, 1.
x"t =8cos30 -6cosg
_-
(1 +cosO)sin0 -* a
xo

2(t+ 3, 1
_-
cos0) A
+'"*xs =2 (4cos8o - 3 coso)
(1+cosO)sinO-*
=2cosB0
1

= sinO =; *2 u2
2G. qR x=Zcoso oTt{ y=9sined, d
+ sin 0 - sin 30o
49 8l
dml
0=3oo('.'o<e< It 6F[: '.'r=7cosQ
0 =30"
+ xz = 49.cos2 e

22. qr( 2sing =seced, d sinag +cos4e tfi1 qH IFTI dtnZ *2 49cos2o
EFT : 2sin0 = secg = 49= 49
2sin0.cos0=1 x2
1l sin20 - 1 + ; =cos2e""""""'(i)
3d6[,
Y=9sino
sinae +cos40 =(sin26 +cos2Q)2 _,1 sin2 2 g
=) Y2 = 81 sin2 g
y2 8lsin2 o
1 11 < 81 8l
, sinz20 =1- 22
_l

-r-
 y2
= sin2e """"""(ii)
rrfirrt rFr = ,u;P
,t
(il + (ii) = JC4)\o)2

,2 u2
= J625
1=+ * =cos20 +sin20 =1
49 81 =25
22. qR sin 80. = cos e d, d e tFT ruiFI qH HI Ettn-
esdqrrH--JP;P
Fl : '.' sin 30o = cos 0 = - J625
1 =-25
cosO
= ;=
cos 60o = cos0
s2. qR f*lfi qtur +r rrq + d d ys stur sT qFr ffi d ara
= Ht-
0 =60o
EFt : '.' ,, tlgqr
28.2cos6 =x+ Ix ,d 2cos2o=? 3n.^
= 18oo

3
U {fSH = U
x180'=108o

EIT : '.'
33. 4 co*
[+ sec2f -.ir,z i =t
= EFt : 4cot26O" + sec230o - sinz4b'

=-(#)'.(#)' (;)
141
=4x 3' 3-2
=,li(r.i4) _q 1_13
326
=;(*.#) ., 34. ?TR a2+b2+c2=ab+bc+cad, d ,s+bs+c3 ET rIFI EFII

29. (sing *cose)t5,1 rfEiIrT qH nfd q;t- *nz

rffiH E?t: a3+b3+c3-3abc=
Ef, : sin 0 * cos 0 t5'T rTFI - ,lW = J, (a +b + c) (az +bz + c2 - ab - bc- ca)
t:
30. lz+rlz+Jz+2cos80 = ? = [ (a2 + b2 + c2) - (ab + bc + ca)] ( a + b + c)
dHr, az +bz + c2 =ab + bc + ca (R+r gur tl
aB + 5a * c3- Sabc = (a + b + c) x 0
a3+b3+c3=Babc
2+z(zalsz+yt) [ '.' cos 20 = 2 cosz e -1] 35. sin Q tFI HFT sec e + 5q { ffi-
EFt : '.' sin26 + cos20 = 1

+ sinze =f -coszg
l2+J2+zcos+e =r-
= sin2e
#
:+ sinO - !'
[-r- sec2e

sinQ
[*,e-1
- !t_ sec2o
=

sinQ - J*Jo-1
31. 7 cos g + 24 sin 0 til rfEifIT iT?It ql{q qH {fd qi't-
36. sin2 B8o + cos2 g8o t[T q;T Hr drn?
EFt : a='l,b=24
6Ft:'.'sin26+cos20=1 1

... sinz3So + cos238o = 1

40. ffi rin20 =
i d, d cos (zb" - e) sr rtt;T IFIT dtnZ
37. secag - sec2g ffi q{r{{ +? ISSC tier I 2012]
ISSC tier I 2012J 1
6FT: secaO-sec20 6cT : sin20 =7
= sec20 (sec20 - t) sin20 = sin3Oo
= (1 +tan20) (1 +tan20-1) {'.' sec20 = 1* tan20} 20 = 30'
= (1 + tan20) (tan20) '. e=15o
= tan20 + tan40 W,
3s. qR sin0+cos0 - J, cos(90o-e)d, d cotefsiln d'nZ cos (zb" - 0)
[ssc tier I 2012
= cos (zb'- t5')
EFt : sing + cos 0 = J, cosO (gO" _ 0) = cos 60o
+ sinO + cos0 = .[ sinO ['.'cos (90'- 0) = sinO] 1

2
= .o.0 = ..8 sin0 - sinO 4I,. fr , sin30 +ycos30 = sinQ . cosQiTgil x sinQ = ycos0, sinO
= coso = (Jr-r)si.,o * o, cos0 # o d, d xz + yz tdilil dtnZ
BSC tier I 20121
coso_tJr_\
= sin0 -\ EFt : xsin3O + ycos3O = sinO . cosO
+ ft sinO) sin20 + (ycosO) cos20 = sinO . cos 0
cot0 = Ji _t
s9. qR psino = iTstt p coso = ld, d ptgl qp4 t+-ilfl dtnZ = G sin0) sin20 + k sinO) cos20 = sinO . cos0
!6 ['.'vcos0=xsinO]
[sSC tier I 2012] G sinO) . (sin20 + cos20) = sinO . cos 0
.... (i)
=
EGI : P sinO = !6 + x sin0 = sinO . cos 0
P cos0 = | .........................(iil sin 0.cos 0
qfrs{q (il 1+, + Y=
sin0
t; .tr = cos e ................. (i)
D- v'J
..... (iiil
sino frI, -=COSU
qfrfi{q (iil t, ycos0 + x sinO

= 0
y cosO = sinO . cos 1qfi51u1 (il tI
D_ 1
r =------: ..... (iv)
cos 0 cos0.sin0 .
q+fi{q + y= aore =srnu
(iiil dstr (i.,r) t,
y = sin0 .......(iil
Jd1
.i"e -.".e
(il2 + (iil2 =
,' + y'= (cos0)2 + (sin0)2 = 1

---\
sin0 \oE 42. qR cosA + cos2A = 1 d, d sinzA + sin4A tFI qFT {rd
=
cos 0
- otFeqr
= tanO = rE [sSC tier I 2012J

= tanO = tan60o EFt : cosA + coszA - 1 ................. (i)

0=60o + cosA=1-cos2A
ffir e .il qFr {dq'{q (iil { {qi qr, cosA= sin2A t'.' 1-cos2A=sin2Al
' sinzA + sinaA
rD_
1
--
cos0 = cosA + (sinzA)'

1 = cosA + cos2A
,--.o.60o -l-, = 1 [qfr*.-{q (il t
'
2
43. qR 4x = ses0 ?TstrT
Ix = tanO d, d cFI qFI cFtrI
 d'nz EFt: secO-cosec0=0
[SSC tier I 2012J + secg = cosecg
EFt : Eqftr qrrd t to 0=45o
sec20-tan20=1 3lt[, secO * cosec0
( q,\2 = sec 45o * cosec 4bo
Qiz-l:l=l
Ix/ tsfi tl = Ji* Ji =2J,
46. qR o (.fi' fil-irf.F {a$iq Ei si}{ 2 sin o + 15 cos2 cr = 7
:3
-x'
1ox2 -{ = 1
d, d cot o qT qH fsdrT drnZ
ISSC tier I 2012]
EFI: 2sincr*15cos2a=7
sino + 15(1 - sin2 a) = 7
2 [...cos2cr = 1- sin2cr ]
2sino +lb-15sinzc--7=o
(il
+ 2 sincr - 15 sin2cr * 8 = 0
15sin2c- -2sina -8=0
15sin2o -12sinq, +10sina -8=0
= Ssino (5sincr -4)+2(5sincr -4)=0
(3sinu +2) (ssincr -4)=0
I =1
=3' 162
Ssina -4=0
tqfr+{q (il tl twilFo Bsin cr + 2 rFT qn (-ve) *nl
44. 2sin20 + 3cos20 HI qftfrT IIFI Hr drn? + 5sinu =4
[SSC tier I z0t2J . sin u:!
6et :... 2sin20+Bcos20 5
= 2sin20 + B(1 - sin20) W,
= 2sin20*B'Bsin20
= 3 - sin20 BC=.ffigFTBF
sinO H'r oTRr*-Td q;T 1 *or tr
3- sin20 sT qrilq rFI =
3- sin20
=g-(t)2 cot cr, : BC3
-2 AB4
45. qR secO - cosec0 = 0 d, d (secO + cosecO) rFT IIFI Vn *tr
fssc tier I 2012J
 * s @!ii?q tti!+r6F*r{f*"r6F*@1s8pt1€€*a*&q}!!$rrigi,ls.1iaf iiJiiirrr: ( ,::t, i3{j{n6: *&4l*#?"*w x

(o+B \ oitrqr+ slur +, d 0 $'r qFr HT Etrn ? IFCI 2012J

1. fr sin26s +sinzF =2€i, d cos
[ 2J =? (a) 5o (b) 90
(c) L2" (d) 15o
ISSC tier II 2011']
(a) 1 (b) -1 g. fr 1.
(c) o (d) 0.5 cos20 -sin2e =5 €1, dt (cosa g
- sina g +1) HI ql-{

n3n5n7n 9n nrd Ht- [pct zotz]

cot cot cot cot cot^=? 1
^ ^ ^ ^ (a) 1 (b)
[ssc tier II 2011] 3
4 5
(a) -1
(c) o
(b)
(d) 1
i (c)
E
(d)
E

t7 10. qR cos43o = tan 47' ih'I rTFT .FTI Etrff?

TR sin6 +cosQ = -\ sinO
e < 90'E), ?lI -cosO -?
G'o< I ru 2012]
BSC tier II 20111 x
(a)
(a) 53
(b)
(b) : v
i 1e
v
(c) 77 (d) (c)
tlx'
r-;------
+ Y'
(a) ,
10 G
qR t.ttQ.tan20 =1€), d sin22 g +tanL1et5l rTFT dkT sinO+cos0
11. q{q
sin0--cos0 =
7 E), d tr.r 0 6T rIFI FII glm t
6t- ISSC tier II 2011J
I ru 2ot2]
(a) 310 (b)
4 ? (a) 24 (b)
d g

(c) 3i (d)
(c) 15
3
(d)
s B

5. qR e qfi +tEqr*, Htur + 3ft1 cosr=+; +, 12. 2sin2g + Bcos2g etg{rl qH fTfird- tcpo 20111
(a) o (b) 3
d cot (90" - e) ot qq nK s't- tFCI 2ot2J (c) 2 (d) 1
13. tan4o. tan43o. tan47o. tan 86o = ? [CpO 2011]
,\ 2J8
(a) (b) 8 (a) 2 (b) B
* G (c) 1 (d) 4
. \ J,
(c) (d) sJ-z 14. qR tanO + cot0
ffi =2(o < o < 90")Ei, d sino -?
i L7
[CPo 2011]
6. fr sec2g +tanz o = *E), d secag -tan40 = ? (a)
2
(b) J5
IFCI 2012J
G 2
1
(c) (d)
(a)
7
(b) ;1 , 1

t2 z lb. fr co.ec B9o =, €), d

5
(c) (d) 1
1 o--^ o
+ sin239o + tan2 51o^ -
1
t5'l rTFT
L2 -
.o.".2*
7. qfE cosx*cosy=2E), d sin-r + siny tfi'l qFT f{*,fd- HT ETITT? [cPo 20llJ
I FCr 2012]
(a) 0 (b) 1
(a) ,t/j 6) l-7
(c) 2 -1 (il (c) (d) t-
*2 -l x2
8. qR tan(29 + 45') =cot3e,
"r6Y
(Ze +45g3Tt{ 3e qfr',
 i:,:;::,:,::,r:::i::ir:i:iiir:ri:;ii:i::::i::iriiiti:iii:iiiiiiiiiiiiliiiiiii:iiiliiliiliiilii:iiiiiliiiiiililii:iiiijiiililli

sin39o
16. ."rbf * 2tan11o tan3lo tan45o tan59o tan79o- 3 (sin221" 25. qR tan 0 =t d, d
+ sin269o) iFT rTFT dril 6,i- [l,oc 2011]
(a) 2 (b) -1
(c) 1 (a)0 (a) 2
12. qR cos2o + coszp = 2 d, d tarrs 61 + sins p t6l rIFI iliT
st- ILDC 2011] (c) 3
(a) -1 (b) 0
26. (secA - cosA)z * (cosec A - sinA )2 - (cotA
- tanA)z t5l q-1
1
(c) (d) Hr drn? [t oc 2011]
1
E 1
18. qR A= tanllo tant29o,B= 2 cot61o cot79od, d F{i' t +{ (a) o G)
2
€r ss *? [roc zott] (c) 1 (d) 2
(a) A= 28 (b) A = -2B
(c) 2A= B (d) 2A= -B 22. qR 0 qs' fro qiur t aqr 7sin2 e t3cos20=4,ti,d g
t['I ql-{ Hr drn? ILDC 2ot1]
B+{ oo< e <good, d g ttrl HFT (a) .6 (b) 1

Is
(b)
[LDC 2011] (c) 1 (d -o
450
28. sin2lo + sin25o + sin29o + .......... + sin2ggo 6"t qH {rd 6t-
(d) r{i't Eti Tfr
ILDC 2011]
20. qR A=sin2g+cos4e,e + q$ qH * ff,q t, re A?FT
qH {rf, s't- ILDC 2011J (a) ,r; (b) trJ,
(a) L< A< 2 (b) 9 <A< 1
(c) (d) 11
4 11
A
(c) 13
(d) 313 l- .-^^ e^^^ 1 I
-<A<1
16- i <A < * 29. cot18" *r sqrr** *o
21. sinzbo + sin210o + sinzlbo + ......+ sinzgoo t6T q;T ila 6t- lcofi2" "o"2zz'* ,*ulpu*]
ILDC 2011] f+-f,rT drnZ 1LDC 2ot1J

(a) 7;
'l
(b) 8; 1
(a) 1 $) J'
1
(c) 3 (d)
(c) e (d) ,i 30. cot109. cot2Oo. cot60o. cot70o. cot8Oo =?
J6
(a) (b) -1
22. fr tan2Q .tan40 = 1 d, d tan3o$T qH {K t5.i- 1

[t oc zollJ (c) .F 1
(d) -T
(a) 6 (b) o J3
31. qR sin Q + cosec =2d, fr ,lrrrooe +cosecrooe iFI qFI
(c) 1 (d)
+ Hr drn?
(a) 1
Q

(b) 2
4 \ \ 3sin0+2cos0 (c) 3 (d)
28. qR tano =; Et, d ffi HT qFT Hr drn?
100

I t oc 2011] 32. qR sec2g +tan2, = * d, d .""4Q -tanag =?

(a) o.b (b) -o.b
(c) 3.0 (d) -B.o 7 I
(a) (b)
24. cos26 +cos40 =ld, d tan29 +tan4e fi1 TTFT.FII trfrnZ n ,
ll,oc 20111 5
(c) (d)
(a) 3
(b) 1
i 1

, s3. qR cosx *cosY=zd, d sin.x * siny - ?

1 (a) o (b) 1
(c) (c) (d) -1
, 2
84. qfq tanA=-; qh tanB = -:6r, dr A+B=? - cos4 [;] .'*'
fi 3n
(a) (b)
, T
6n
(c)
4
(d) fli't 6ti Tfr' (c)
*
(d)
16
3

qR sin (A + B) =
l.rA 46. fr .".9 + tan O = 4d, d sin g t5T TTFI FTfiTd_
35.
,3it{ cos(A-B)=181, Hl ;o-r ffi 15
(a)
Hr drn?
(a) 1 (b) z
*
(b)
i
8
(c) J, (d) .6 (c) -G 8
(d)
i
cotA+tanB
$T IIFT q?Tr Etrn? tan0 tan0 ERI.R r\
ffi 47.
sece-l
*
.""g*r E-
(a) cotB tanB (b) cotA tanB
(a) 2cosec Q (b) 2sec 0
(c) cotB tanA (d) {rd rfr f*q qr {FFdT
(c) 2tanQ (d) 2cot0
37. fr cosg +sino =ad, d coso -sinQHT gFT HI dtnZ
( r \
(a) J;7 G) JR tan Q t;A,,1 trT qFT t{frrd'-
(c) J;J (d) J;? (a) e (1 (b) 0 (tan'o - r)
cot - tan2 e ) cot
38, tan72O" - cos270o - sin150o. cos120" qtffl +- (c) cot 0 . tan2 g (il tan0. cosec2g
(a) (b) I 1+tan20
: i 49. _________6_
1+cot"0
=

I
(c) (d)
i
qR sinQ +cosec0 =2d, d sinze +cosec2o qfffl +-
(a) tan2 e (b) (##)'
39.
(a) 1 (b) o (1+tane \2
(c) z (d)s (c) (d) [r-tane)2
t r-*e.l \1+tanO/
40. cos3x = t d x i5I HFI Hf drff ? g)=
cos (A + B) cos (A -
(a) oo (b) 45o (a) sinzA - sin2B (b) cos2A - sin2B
(c) 60o (d) 9oo
41. sin (A + B) . sin (A -B) = (c) cos2A - cos2B (d) cos2A. sin2A
(a) sin (A2 - 82) (b) sin2A 51. qR e+B+c=lgo'd, d tanA*tanB+tanC=
(c) sinzA - cos2B (d) sinzB -sinzA (a) Ztan Atan B tan C (b) tanA tan B tan C
qR
(c) cotA cot B cot C (d) 3tanA tanB tanC
42. cose +./3sin0 =2d, d gtfi"T rfl;T dfd St-
52. qf( .fr cos g + sin e =1 d, d e Hl rrFT +rT drnZ
It 2n
(a) (b)
5 T (a) o (b)
It
3
4n
(c)
T (d)7 5n
(c)
; (uJ
;
.o I
53. cos 0 - sin e t5'T qfiFI qH .HI dtnZ
4s. qR srn'0 = *t o< e < n E1, d e+T fsdqT qH drn?
7
(a) 1 (b) 2 (a) J, (b) 1
(c) 3 (d) 4 I 1
(c) (d)
44. B tanz4f'- sinz6o" - rIH Fr6rd- , O
! "otzeo'* * sec245o a.T
54. sin 105o . sin75o E5'l qH drr-
(a) 1 (b) 2
(c) 3 (d) 4
(a)
+ (b) v3+z
t7

2
 * *(,*!i{:}Btxilx:lxffi{*$3**$#*Er:ijn$ weeffiliffi*1s#*#l*}Lls#ir*w
(c) J, (d) 2 J'
64. fr secg+tan0=Pd d singe{-q{ }- (LDC2012)

bb. qR tan 0 - J, d, d e cnl qtr Fffird -

lt
(a)
it* (b)
i+*.
(c) ltf +*ufi.7r
(d) t"6
f cot3Ol -cot751
56. tan5Oo. tan4Oo .['I rIFT UfC H.t- tan150 -tan600 =rl
rTFT drn- &DC 2012)
(a) o (b) 1 (a) -1 6) o
(c) z (d) E{i't Et$ Tfr (c) 1 (d) ,t3 - t
57. sin6 a * cos6 6x + 3sin2 c, cos2 o tET HFT nfd t5.t-
cot6. tan (900- (900- e). Cosec0 + (sin2250*
(a) o (b) 1 e)- Sec sin2

1
650) + .,/3 (tan 50. tan 150. tan300. tan 750. tan 850) +t qft
(c) (d)
t- (LDC 2012)
,
58. sino g + cos6 e cfi'I rr6\H rTH qqT d'nZ (a) o (b) 1
(c) -1 (il2
(a) I (b) 67. fr cos e .cosec 230 = 1, d etnT qFT t : ODC 2012)
i (a) (b)
(c) 1 (d) 2
2Bo szo
qR (c) 680 (il 670
rg. cr+ 0 =gg"eftq a=28 d,d cos2 cr+sinzp E-r rTFI
68. sin250 + sin2100 + sin2150 + .......+ sin2850 + sin2900 .E'T qH
{rd H,i-
6:
\ (LDC 2012)
(a) (b) 1
1
, (rl t, 1
(b) a;1
(c) o (d)2
60. (tan g t sec e )2 qfrqf *-
1*cos0
(c) Lo,1 (d) ,i
(a)
1{ose
zTI(tanl(; I

ril;T Hr dqr-
1+sinO Z)=..6E),d "o.66T
(c) (t oc
l-sme 2012)
( t r \( 1 - 1 \ tFT {r€qrirrir rlFI fsffir+l
61'
[.".e.."oe] [."* "t g)
0 G)6 1

(LDC 2012) 1
(c) (d) 1
(a) o G) -1 2
(c) r (d) 2 2sin0-cos0
zo. qR
cos0+sin0
- t d, d cotgthT IFT wr drnZ
62. qR
H#.j*=;d,n ##6r rrFr tutr{r d,nz
(t oc zolz)
(LDC 2012)

(a) 25
(b) 4le
(a)
1

2
(b)
(d)
i
G (c) 3 2
zr. .rR cose+sece=2d, d cos66*sec6gq r11;I t :
(c) 4l (d) 40 (LDC 2012)
40 4t
(a) 4 (b) 8
(c) (d) 2
68. qR sins-cose=0,d sin(;-r)rcos (;-r) "il qH drn 1
qRa ABC I
72. zl,= 800, 2B=600 d, d Tfr qrq d zcitiT
oiz (LDC 2012) q,tr (LDC 2012)
(a) o (b) 1
 ii:ii::iiiilr:ii::iiri:ii:ri:::i::i::ii:ii:::i:ir::::::i:::i::::::.::::li:l.l,l:::li:li:l

t;
(a)
T
%f
(b) + (c)
=2
(d) 1

(c)+ (al+ 81. qR o (eF' ffifeffi {"iuf d nqf 2 sin cr

d cot cr tFT HFT t*,ffiI dtnZ
15cos2* = 7
@oC 2012)
d,

52
73.;pe*,*;7;+ssinzsH'l q€qkrfi qH fsil{r *ni (a) 34 (b)
4 B

(LDC 2012)
(c) 62 \o/
(d)
(a) 2 (b) 3 =z .6
(c) 4 (a)b fr (a' bz) sin 0 + 2ab cos0 = az +b2 d, d tan 0 t5t rTFT

74. qR xsin3O + ycos3 0 = sinO cos0 ?t2jlT xsinO = ycos0, Hr drn? (uoc 2012)
sinO, cos * 0 d, d f + y' f{dil dtn? @DC 2012)
(a) (b) Lru'-o',
(a)
1
(b) 1 fio'*a'l
Tz ,
(c) (d) Lr@'*t')
(c) 1 (d) J' fio'-a't
75. 0* tan 0 * sec e) (r + cot 0- cos 0) t6.f v{frSd qH *- gs. qR tan0_cot0=adetT cos0_sin0=bd, d 62++)(bz
(uoc 2012) - 1)2 E-r q-{ EFrr *nZ (uDC 2012)
(a) 4 (b) 1
(a) -2 $)z (c) 2 (d) 3
(c) 1 (d) -1
76. qR sinz cr = cosr cr d (coto a - cotz cr) qt qI-;[ + l-sinzAT--l+cosA sinA YI \ ^
ot. Str.a', rt't Y-"11 Qt'['
(uDC 2012) 1+ cosA sinA 1- cosA
(a) 1 (b) 0 (uDC 2012)
(c) -1 (d) 2 (a)cos A (b) o
(c) 1 (d) sin A
+ @DC 2o1z)
85. 4 cosec2 cr * 9 sin2 cr 6'f qliIrl qH flEFdTf *nZ
1-sin0 (uDC 2012)
cos 0 (a) T4 (b) 10
(c) 11 (d) L2
1+cos0
sin0
s6. qR cotArcosecA=3d sil{ Aq.F qr$tuT d, d cosAtFI
TTI"T ff,flfl d'nz (uDC 2012)

zs. ?TR
#.#=1,d #.#* qH +
(a) +
D
(b) 1

(uDC 2012)
3
(a) 4 (b) 0 (c) (d) i
;
(c) (d) 1
87.
:
tkfi qrq (uDC 2012)
2.6ffi.tr r<fsn, fr AB-BC=2tfr d, d (cos2A-
79. 630 14' 51" t5',t
cos2C) tFT rTFT Hr drn? (uDC 2012)

(a) (fl*')" (a) ?

D
(b) I
D

(c) (*H.)' (c) I

D
(d)
*
88. qR A iTeTI B T(+. Etq EI, d sinA cos B + cosA sinB -
80. fr "e"0+tane= J$,d tanAtanB + Sec2A - cot2B tfi"t qH tddqT dtnZ
UDC 2012)
(a) o (b) , 1 (a) 2 G)o
(c) 1 (d) -1
 liiiirriiilr,.:.l:,::l:lririiri:ilrii.:'::#iiiiliii ii,!*

89. qR 2- cos20 = Ssin 0.cos0, sin0 + cos 0d, d tan 0tfiilfl I-;
r/1-a" +1
d'nz (uDC 2012)
1
(b,t
lL7
(a) (b) 0
,
(d)
(c)
?
(d* qI;I fiF-im dtnZ
98. tan 9o.tan 27".tan 63o.tan81o tf,'[
90. qs' vtrmiur frgu ABC d, AB = 2.b tfr, CosB = 0.b, (uDC 2012)
IACB= 90o Tqr AC 61 ffi, .r" d tt @DC 2012)
1
(a) J5 (b)
(a) G
i,tA
1
(c) (d) 1
(c) ,
bJ5

4 . ee. qR o<e.;d furh { ,".0+tan, =;d,

91. fr cos0 =f ,d
cosec0
ffiHT qH + @DC 2o1z)
d ,". 0-tan0 trl rIFT t*-il{r dtnZ @DC 2012)
5 4
(a) (b)
i 7 (a) ?
b
(b)
i
(c) !
D
(d) i (c)
;
(d) 3
i
92. 0m qn [0"<e<eoo],ffi ftq #*ffi:a *ffi
100. zTR /cos20 + m sin20 = ,oo< g < 90o,d
(uDC 2012)
(uoc 2012)
(a) 4bo (b) 60'
(c) Boo (d) E{i't 6ti Tfr
93. qR cos0rsec0= J5,cos30+secse6'f qH +
(uDC 2012)
(a) 6) J5 101. 0* qfi qrwk*. rTrn + tclq sins 0 + cosra eHr 3TF+5ilI
(c) 0 (d) 1
qH+ (uDC 2012)
94. fr ,r. = coso cr * sin" od, d 2r. - Sun * r f+-rtr *nl
(uDC 2012) 1
(a) (b) 0
(a) 1 (b) O
(c) 6 (d) 0 (c) 1 (d) Ji
fr sin 0 + cosec O = 2,d sin?0 * cosecT0 6f rTH t
(uDC 2012) 102. qR o<e.;d fr"{fr d s""Q+cos e=Zd, d seco
(a) 2 (b) 0
- cosO ?5T qT;T fciilrT dmZ @DC 2012)
(c) (d) T 3
1
2 (a) (b)
, z
n rc 3n 5n
_tafl-tan-tan- ,fi (uDC 2012)
96. fqn
"-'^8 sln-
t2 8 t2 - -6 (c)
;
(d) ln

1
(a)
4 sin 0 tF'I rIFI ii5,dil frnZ
1 (uDC 2012)
(c)
2
65
e7. qR sin(10"6' 82") = u, d .o. (Zg"b3'2g") + tan (10"6'82,')
(b) 144
r6't qH+ (uDC 2012)
 209 25 5n
(c) (d) loe
(a)
T (b) , TE

7s

.
ror. 4qo tan57o+cot37" ^
iffi.*ff#f{rs$
\
q{r+{ t? @DC 2012) (.)
+ (d) It
8
(a) tan 33o cot 53o (b) tan 53o cot 37o
(c) tan 33o cot 57o (d) tan 57o cot 37o 1t2. fr ta., ,rr"=rd, d co, ur;'$T rrFT Hr drn?
(FCr 2012)
r,E. fr 2 sin (T)= ,'*$d,d (,-i)* qn t+rn d.nz
(a)
(uDC zotz)
(a) 1 (b) 0
(c)
-1 (il2 (c)

qf( Cosx= SinyiTslT cot G- d,

id, n
106. qR sinl70= (secl7'- sin?3')tfi'I rTFT fs,Hil dtnZ 113. 40") = tan (50'-y) d,
(uDC zotz)
iISTI y tFT II;T Hf dqr? (FCr 2012)
(a) 70",20" (b) 8bo, bo
v' (c) 800, 100 (d) 60o, goo
(a) (b)
*'-y' LL4. qfq tano + coto = 2 d, d tan1000 + cot1000 f6dTr dtnZ
IFCI Mains 2013]
(a) 1 (b) .6
(c) 2 (d) 0

102. qR eq*. wtrirs' =1r+.tur d, ott tan 20.tan B0 = 1d, d llb. q+ qqfrqe hgv *
qn qlur ar Afra rrv
T *r
z"or'!9
z -,'J* * d'nu
(
fsilrT @Dc zotz) il{5qR Es+ eaq 6M + Tfrq qrq wr diz
t
[PCt Mains 20fi]
(a) o (b)
; (a) 2n
(b) 4ne
9
(c) (d) 1 5n
; (c) 18
(d) 5ne
ros. .{R Zn aen lg qqsq 1r+ aiu d, d
sec2 A * sec2 B - Secz A. Sec2 B t["1 qH fd5,drT ilnZ qR secO=x+n (0'<e<90')d,
116. d secg+taneffi
(uDC 2012)
(a) 2 (b) 0 q{l"R frnZ IFCI Mains 2o17l
(c) 1 (d) -1
(a) x (b)
*
10e. (2cos20-r,(=#.i#) qn tqrn frnz
(c) (d) 2x
fpct zotz] ;
(a) 3 (b) 2
(a) L sec0+tan0 5 \ \
(c) 4 tr'. qn ffi =
s 6t, di sino tahrrd Eruqt 6rrn?
110. qfR 2 (Cos2 0- Sin20) = 1d, eh e trnerer
{l6tuT d, d
e ffi rn;T f6ilfl drnZ (FCr 2012) [uoc 2olsJ

(a) 45o (b) 22L"

2
(a) 3
i
(b)
i
(c) 60' (d) 30' (c) 1
(d)
5 ?
,1t.\ qR usin0+bcosO=.d, d acos0-bsin04'I qFT ti5--dTf
1r1. d EM'q-r frr 13b"+ dsTr sftEr srd{
; Br rqTsn W 118.

*tq *'r tfsfi rTrs Hr frnz (FCr 2012) fuoc 20fi]

(b) tJ;\b\c
-- t6 --
S.S.C. ADVANCED MATHEMATICS
 iriiiriii:i;:liiiiiiiiiiji,iilliiiiii,liiiilii II

(c) 4;\b, - c, (d) tJ;, -b, - c, 1

(b)
11g. qR coso + cos20 = t d, d (sinl2e + 3sin100 + Bsin8o + V,J

sin6e - 1) H-r s€qrrr.F qH ffiT{r dtnZ tuoc 201sJ (d J5

(a) 1 6) -1 128. qR tan0+cotO=2,d tan'0*cotnO (0"<e <90o, ng.5'
(c) 2 (d) *
tqt-6 t> oT qH
0
tuDC 2o1sl
120.?TR x=asec0.cos$, y = bsec0 sin$, z = c tanO Ei, di (a) 2 (b) 2'
x'qTv' z' (c) 2n (d)
a- D-, - c-,
$l 5 ffi cr$R drnZ tuoc 2olsl
129. qR sin (60" - d (y
2n+t
tun (v t: (qn d
e) = cos - B0) - 0)
(a) o (b) 1 fq'eTQT vfr wrerro {fl"iuT t te<60oiTetrt v >80"+
(c) 4 (d) e Hrq) fuoc 2o1s]
121. coslo cos2o cos3o coslTTo cos178o cos179'6[ TII-JT
1
f{il{r +? IUDC 2013J (a) (b) 0
15'
(a) o (b)
1
(c) ..6 (d) 1
,
130. (1 + cotO - cosecO) (1 + tane + secO) H'r rTH +
1
(c) 1 (d)
o (a) 1 G)2
[uDC 2o1s]

22\ (c) o (d) -1

rzz. ttfsfi H'r Bfi i qrq (,= T) trilrT dm tu
181. qR sin(A-B) =l u* cos (A+ B) =i =,ar'A> B > 0rTsTr
[uoc 2o1s]
(a) b7"6r'22" (eqqq) (b) 57oL6'22" (Vqqq) t d e iil rrFT Hr drn? tuoc zotsJ
(c)
s7o22't6" (etw; (il b7o22'L6" (etw; fi
12S. qR sinO + cosec0 = 2 d, d Ht q-{ HI
(b)
sineo + coseceo n
d'nz [uoc 20rc] fi|t
(a) 3 (b) (c) (d)
2 ; ,
(c) 4 (d) 1
132. fr (r+sinB) (t +siny)=(1 -sincr) (r-sinB)
(1 + sino)
124. qR sec0+tari0= z+ Jt d, d sinO + cosO H] q;l IFII (1 siny), d rfu gw ffi q{FR drnZ tuoc 20181
-
d'nz [uDC 2013] (a) + coscr cosB cosy (b) t sincr sinB siny
3
(c) + sinct cosp cosy (d) + sincr sinB cosy
(a) r= (b) .6 1BB. (2sin0 + 3cos0) q'r erfir+cm rfl-{ HI drnl IUDC 201a1
vb
(a) 2 (b) fi5
/\
(c,
7
(d)
1
(c) .E
vb
t=
vb
t= (d) 1
125. (sin225" + sin265') tFT rTFT tsil{r +2 [uDC 2o1a] L34. L52 (sin30" + 2cos245" + 3sin30" + 4cos245o + ............... +
17sin30o + 18cos24b9 sl rIFT iFtt drnZ IUDC 20Bl
(a)
+ G)1 (a) tqt-*. qtg, Wf qrf Tfr
(b) qRtq q@r qtg {.rf-6 Tfr
2
(c) o (d) (c) Wtn ul Wf qrf (d) srqRtq
15
sin0 cosO \ \ 4 ^.o cr1
126. qR ; = i , ET, il, sinO - cosO i6[ qlf * 135.
*.*;.,.;*+3sin2 fi'r qH * IUDC 2013]

IUDC 2OfiJ (a) -1 (b) 2

(a) x-y (b) x+y (c) 3 (d) 4
x-y y-x t
136' 1.;Pe '1+ ,' u . + 2 sin2 o sr qEqrir1fi qH dqr-
(c)W (d)
w tan'e
[UDC 2OfiJ
127. qR sec0 + tan0 = ..6 (o'<e<90")d tanedtnZ (a) 5 (b) 6
IUDC 2013J

.. S.S.C. ADVANCED MATHEMATICS

 I.li ii:i i1:lii:il:ilili:iliti::i;:i:ili1::i::i;ri;::ii:i:::i:::.iij:;i::1t:,i:

(c) 3 (il2 m'-1

1S7. qR sinO + cosec0 = 2, d sinso + cosecso HT qr{ + (c) ___=_
n'-1
(d) 4+
n-+I
[uoc 2ola|
(a) 1 (b) 0 146. fr cos{e . sin4e =;, d r - 2sin2o q,T qH +
(c) 2 (il; fuoc 2013]
4
(a) (b) 0
138.'".r(#.#)-zt",,e sr qH +
B

2
[uoc 2o1s] (c) (d)
d *
(a) 1 (b) 2
(c) o (d) 4 r47. (T) trs{{ ffi q{lq{ +? IUDC 2013J
139. 3(sinx - cosx)a + 6(sinx + cosx)2 + 4 (sin6x + cos6x) .FI TII;T
+ tuDC 2o1sl
(a) 1oo' (b) tzo"
(a) 11 (b) Lz (c) 108' (d) 180"
(c) 18 (a) L4 148. qR tan0 * cotO = 2 d, d tan20 + cot20 t*-f{r *nZ
140. qR e qiriq-fi qrsiur + dsTI B(sec2o + tan20) = b, d [uoc 2o1JJ
+h-qr t{rw +rfi tu tuoc 2o1sl (a) 2 (b) 1
(a) cos20 = sin0 G) cos20 = tanO (c) J, (d) 0
(c) cos20 = cosO (d) cos20 = sin20 149. 3cos80o coseclOo * 2cos59ocosec3lo tFT rT'FT fn +tHqt
11.\ tFI rIFI B [unc zolyJ
141.'
;. * - ."* - r*, [uoc 2o1B]
(a) 1 (b) 3
(a) 1 (b) cotO (c) 2 (d) 5
(c) cosecO (d) tanO
I co.' A(sinA +cosA) sin2A(sinA -cosA)l
142. qR coso + sino = .E cos0, d cosO - sinO t 15o.L.o,".,nGi,oa_.o.e)+ffij tsec2e-
[uoc 2013] t[T qFT f+m tz
cosec2A) [uDC 2o1s]
(a) J2tano (b) -J2coso (a) 1 (b) 3
(c) -.Esine (d) J2sino (c) 2 (d) 4
151. xcos0 - ysinO = 2 dsfi xsinO * ycosO = 4 i g H'f fT{ffi{ufcE'e
14s. zTR tan0=9 iTsTI eqrstur +, d cosecet d wr yrw drlr? [uoc zo13]
tUoC 20Bl (a) x2 + yz - 20 (b) 3xz + yz = 20
(c) x2-y2=20 (d) 3x2-y2=10
4
(a)
=b
(b)
; 1b2. sin20-Bsino t2=0frq { t trs furfr { vfr frnZ
[uDC 2o1s]
(c)
i (d)
t (a) o<e<90
(c) o=oo
(b) o<e<90
(d) 0 = 90o
[uDC 2013]
153. cosax + cos2x =t d, d tan4x + tan2xH'l IIFT Hf drff ?
[cPo 2013]

(a) 1 (b)
1
Z 2

(c) (d) 1 (c) 1 (d) 2

2
, 184. fr 7sin20 + Bcos2g = 4 d, d coso (0" < e < go') tFI rTFI
145. qR tancr = n tanP dell srno = m srnp-\o\rll cos"cr 6 tsilfl drnz rcPO 2013J
[uDC 2o1a]

(a) --m'
n- +I
(b)
m'
n-
(a)
+ G)
J5
2
r=
(c) vb
1 (d)
T
15b. 4tanze + gcot2g +l {iiFI qH HI dtnZ ICPO 201sJ
(c) 45o

(a) 13 (b) L2
(c) 1 (d) 6 164. ?TR 5 tan0 = a t, d
156. qR sino-coso=
| a, d sino*cosoHl qt;t fdilrT dtnZ
fcpo 2o1s]
(a) 1

Z
(a) 2 G) -2
(c) 1

(c) +2 (d) h E
2

157. qR coseco - cote =f, d, d coseco tR'I IIFT Hf drff?

168. ?TR.rsing-ycoso = Jx, +y, .ih ffP.#U =#
ILDC 2o1s]
[cPo 2013]
(b)
(a) i49 (u) 47
,,
#.5=,
(c) 51 (d) 53 (d) *-4
D-
=l
a-
* 28
166. qR 2 (cos2o _ sin2o) = 1 (e qrfrtur *) d coto tahrrd

1b8. zTR x sin4bo = y cosecao'd, di fr*q qttn trnZ qnfl +? ILDC 2013]
1
IUDC Mains 2013] (a) (b) 1
15.
(a) 43 (b) 63
(c) 23 (a) 83 (c) (d) _.6
^6
L67 . 2cosec223" .cotz 67o -sin22 go
- sin267o - cot26 z" ffi ERFrt
159. qR rsing= 1, rcosg= .,8 d, d (.,6tane+r)) $"t rIFT tz [t oc 20fi]
Hr drn? [UoC twains 2015] (a) sec223o (b) tan233o
(c) o (d) 1
(a) .6 (b) /,t,
(c) 1 (d) 2 168. qR cos43o=
fu t, d tan 47o+'r qH +

1G0. (0':o.e0)81 tuTh { J.o*-,_.o* tndn ar.nz ILDC 20fi]

[UOC Mains 2013] (a) +
(a) 2 cosecA (b) 2 secA
(c) 2 sinA (d) 2 cosA
(c)
161. ?TR sing + cos0 = .,,E cose d, d (coso
- sino) t['T rTr;T

f+-dar drnZ IUOC tWains 2013]

(a) .6cosO (b).6sino
(c) .Ecose (d) JZsin0 ILDC 2o1B]
162. gfi Irqer6zuq H,T tr+d, sTrilR-fr 6iq ffi ifsT{ 6,T +dr (b)
tZ [uoc tutains 2o1BJ ;
(il1
(a) % (b) 3%
sec0+tan0 Z+JZ s r
*.e_t.r"e =r4 €, dl {
(c) 2% 170. zIrE Tfr qrq e ?fiT rtFT
(d) %
16s. 96 qrfrtur t*gg {, t
Tqret sT Uur+$tr ffi Tqr dtn- ILDC 2013]
(s{rtt( mril * erf mr enqr tr ilq1qR ysFT l1fi a-{*i"r fi
(a) (b)
tddqT *+ qrftqz IUDC Mains 2013] i ;
(a) 60" (b) Boo
 : t!!rii:::r i::iitrli:::rir!r!!lr|!:rlr:rir:rir!:lr|!l !:, i:rl:ii:;ri:
it,itr::iiiiiii::rt:it:iii::i::ii:i::iii::i.::iiii:iii:ii:iii:ii:ii:iiiii:ii:iii:ii:iii:i::ii:ii:iii:ii:iiiiiii:li:li:iir :il: i

(c) ;
fi
(il; tan2B) th-t qn lqrqr d'nZ c 2ofil
b (a) 1 (b) -1
(c) o (d) 2
121. qR seco-"o.e :| (eq+ tll {r+tur tl a seceffi 179. qR tan0 + cotO = 2 d, d sin0 + cos0 "5t qn Wt q'tf
t
erdrt ILDC 2013]
ILDC 20131
(a) o (b) 1
(a) o (b) 1 (c) J, (d) 2
2
2sin68' 2cotl5" 3tan45'.tan20o.tan40".tan70o
(c) 2 (d) -1 180' *"2t - 5t"."?5'- 5
2

qR
ffiqrg{*? [t oc 2olgJ
172. tan22L" = x t, d cos67]" ffi qrrq< tZ (a) -1 (b) 0
(c) 1 (d) 2
[toc 2o1s]
181. coslo cos2o cos3o cos4o cos100otil UuFilFe ffi
(a) x (b) ,h.., qrr{{ +? [t oc 2o1sJ
Lx (a) 1 (b) o
(c) (d)
G G (c) -1 (d) 1

173. sin2lo + sin23o + sin2so + ............+sin287o + sin2Sgo 6-l i

qrr fs'il{r dtnZ ILDC 2ol\l 182. sina0 + cosaO 6'[ eTRrdrilq qH 16il{r drnZ ttoc zofil
(a) 22 G) (a) 1 (b) 1
"; , t

(c) 2 (a)3
(c) 28 (d ,r; 183. zIR 3sin0 * 5cos0 = 5 d, d s .ine - 3 coso f+:FS tRFn t

L74. cos20 + sec20 Er qlffi qrl flsil{r dtnZ tloc 201s1 d'nz [t oc 2o1s]
(a) (b) 1 (a) 1 (b) +2 1

o
(c) (d) 3 (c) r B (d) +b
2 1

184. fr sinO+sin20=td, d cos2e + cosao tFI rTFI tfr.il{r

175. qR cosO+secO=2(0'<e< 909*, d cos100rsec1106FI
qH f*,T{r drnZ d'nz [LDC 2o1s]
ILDC 2o1s]
(a) (b) 1 (a) o (b) 1
o
(c) (d) -1 (c) 2 (d) 4
2
d qn f+fqr 185. tan10'tan15" tan75" tan8Oo tF1 rTFT tfif,il dtn2
176. zTR secO - cosecO = 0 d, tan0 + cotO iF'l
[LDC 2013J
frnz ILDC 2o1s]
(a) (b) 1 (a) 2 (b) 0
o
(c) -1 (d) 2 (c) 1 (d) -1
186. 4tan20 + gcotze fir qrcq qrq ffi qtFR drnZ
177. cosec:l8o =#u.h'T rTFT tf,ilfl ErrTr? ILDC zols] ILDC 2o1B]
(a) 13 (b) 0
(c) (d) 12
(a)
15.
1
G)+ 187.
5
fr sin7x=coslkd, d tangx*cot9xt5'T qH flSil{f *nU
1 [LDC 2013]
(c) (a)
,
1
(a) 4 (b) 1
178. qR o+B=90od, d (1-sinzo)(f -cos2cr)(t+cot2p)(t+
(c) 2 (d) 3

-r-------
1. sin2 cr + sin2 9 =2 tanQ .tan20 = 1
sin Q <1 1

sincr = sinB - 1
+ tanQ - tan20 =
cot 20

cr= B =90o = tanQ =tan(gO'-Ze)

/o+B\ 0 = 90o -2e
.o.
I z ,J =cot90o=o = 3e =90"
0=30o
Tc 3n 5n 7n 9n
q,
cot cot a cot cot cot n 3lt[, sin2 2 g + tanz 20 = sin2 60o + tan2 GOo

^ cot27" cot45"^cot63o cotSlo

= cot9o ^
= cotgo cot27" cot45o cot(g0o - 2n cot (90" - 9') + (J5)2 = 4
3
*3= -153 D_
4 -o4
-
= cot9o cot27" cot45" tan2'7" tan9o
= cot9o tan9o cot27" tan27" cot4lo 15
=1x1x1=1
t7
3. sin0 *cos0 = 11...........(i)
qfqf fs-sinQ -cosQ =x......(ii)
+n q+f,{q Ei q{ s5$'ff w
/ tt\2
2(sin2Q +cos26)= txL
@8
t;l I __

289
llzsg L7

+ 2x1= ,*+x" 8

289 sin0 i8
x2=2- 169
3lt[, tan e =
.o.e = 15 =G
17

v-
,49
=- 8
169 cot(90-09=tan0= 15
7
':
. Y=
6. * tan2g =
7
-13 sec2g
o
-- 2L --
S.S.C- ADVANCE D MATHEMATICS
 sec4e - tanag
(sec2 e )2 (tan2 g )2
= - x2+y2
= (sec2g - tanzo) (sec2e + tanze)
77 v
='l.L2L2
Y cos47o =
W
--
cosr*cosy=2
t fO cosx*cos ytfi'l rIFI 2drnr t nfr grrq t qe
6lT ql;l-d
x=y=0'dl tan4'7" =
sin47o
ffi, cos 47o

cos0o+cos0o= l+L=2
sinx*siny
0o * sin 0o
= sin
=0+0=0 v
8. tan(29 +45')=cot30 sinO+cos0
11' silgr*org = z
= tan(29 + 45") = tan (gOo- eg)
=:) 2e +45"= 90'-3e = sinQ +cosQ -7sing -7cos0
=) 20 +'3e = 45 + 6sin Q = Scos 0
45" ti rm cos s t qrq ti qt,
o = u =9' 6 tang = 8
9. f<+ ge+r t '
84
tanQ =6=l
cos2g -sin2e = *. ....,, L2. 2sin2g + Bcos2Q
cosa6 - sinae + 1 =2sin2g +a(l -sin2e)
= [(cosz e )2 - (sin2 g )2] +r = 2sin2e + B - Bsin2e
= (cos2g + sinzo ) (coszo - sin2o) + r -3-sin2e
=1x(cos26 -sin20)+f sin 0 i5'T 3Tft{-.5'ilq TTFI = 1

1
3-(t)2= 3-l=2
=1x B *1 tqflfiq (il I 18. tan4" . tan43o . tan47" . tan86o
= tan (90" 86') tan (90'- 47) tan47o . tan86o
-
T4 = cot86o tan86o tan47" cot47"
=5*1=5 =1x1=1
tanO+cotO
L4
tanO--cotO
-2
10.

sinO cos0
*
cos0 sinO
[cos (90" - 47") = sin47'] |
= sin0 cos0 =
cosO sinO

sin2e + cos2o

sinO.cos0

sin2e + cos2o
aod -2
sinoO - cos"0

1
+ ;;% -2
-;Je
1
+ sin2 g- cosz S
2
 1 cor2 e
=-(cos20-sinze)- , 19' [sPe--"oszd =
3

+cos20 =-,
1
' cos2 e
3 cos2e =3
=> cos20 = cos (180'- 609 sin20 -os20
+ 20 = L2Oo -Jt
' co.2 e
0 =60o cos2e-sin2o.cos2o =
= . sin20
B

sin 60'=
Jl
, cos2 e.sin2e
= ;;;zoG-;m, = 3
1^^1
18. .o."Jbf + sin239o + tan2blo - .irr2 bt.lr" cosz e.sin2e
"z,,go
= ;;;2e;;;ze- = 3
, _:_2ono, sin251'
?srrr ov -;;;%f - sin2 b1'x sin2(g0'-b1')
cos
""2s:r" =) tanzg =3 +tang = !6
sinz blo + sin2 (90'- b1') + tanz blo - - = tang = tan60o
5 1o. cos ec 510 0 =60o
= sin2blo + cos2blo + tanzblo - 1 20. A=sin20 +cosao
= 1 + tanz (90'- 399 - 1 + A=sin2e +(1 -sin?e)2
= cot? 39o = cosec2 3go - ! = x2- |
= A=sin20 +1-2sin20 +sin4e
16.
sin39"
+ 2tan11o . tan31o . tan45o . tan59o. tan79o =A=1-sin2g +sinae
- 3
"".bf
(sin221'+ sin269')
= 1-A=sin2e -sinae
+ 1-A=sin2e (1 -sin20)
sin39'
- cos(go"-Bg") 't t2tan11o . tan31o . tan4lo tan (90"
- 31) = 1-A=sin2e.cos2e
tan (90" -119 -3 { sinz 21"+ sin2(90'- z1)}
= 1 -A- (2sinq'cose)2 = ] {.i,ze),
sin39o
- sin39o -t z/o tan11o. tan31o. tan 45" cotSlo cotllo - 3 = 4i-A)=sin22e
(sin221" + cos2 21o)
+ o < 4(t -A) < 1
=
=1*2-3
1+2 tanlL" cotllo tan3lo cot 31o tan 45o - 3 + o<(1-$< i
=Q
17. cos2cr *cos2g=2 =) -1 <-A. - I4
Td drfr q,rq t "lt[ cos2 o ?tql cosz B H-r qH 1 d 3
cos2 cr = r d}t cos2 p -1
<A<1
4

= coso =1=cos0o 21. sinz5o + sin210o + sin2lbo+ ......... + sin24bo + .......sin28bo +

3tR cosp -1=cos0o sin29o'
o =0o&=0o = sin2bo + sinzSbo + sin210o + sin280o + ........ + sinz4bo +
tan3g + sinsp sinzgoo
= (tano )3 + (sinp)E . =1* 1 + ............+ g Eilg.r + sin2450 + sin2g00
= (tanoo)B + (sin0")5 ('t\2
1=91
-
=0*0=0
18. A = tan11o. tan29o ...............(i)
=8*tel +(r)2=8*L*; z
B=2cot61o.cot79o 22. tan2Q.tan4e =1
= 2 cot (90'- 29) . cot (90'-119
= 2tan29o . tan11o
... s = 2a 1sfr+,.qu1 (f)
+ tan2o = #*
3ik[:, 2A = B
= tanlg =- 4
tan(20+20)
 l-tanz20 sin20+cos2e--cosae
tan2o =
zt^nze cos2o

= 2tanz2O =1--tan22O 1<osa0

3 tan2 2e = L cos2o
1
+ tanazg =
3
1-sin20
tHfio (il1
cos'0
1

= tan20 =15. aor2e

-
=--a
.o.2e
:+ tan20=tan30" Shortcut :
2e =3oo cosz6+cos40= 1
0 =15o cosae = 1- cos20 = sinze
tan30 =tan45'=1 tanzg + tan40
Shortcut :

L-tan2 0. tan 4e =0 sin2 e sina e

- ' l-

.o.2 e cos4 e
tan20 + tan40
+ tan(20 +40)= I - tan 20.tan40 .o.4 e sina e
+ tan60 = o - .or2 o
' t_

sin2 o
+ 6e =90' = cos2 6 + sin2 e
0=15o -1
tan 30 = tan 45o =L 25. tanQ =1=tan45"
3sinO+2cos0
23. 8sin0+Scos0
3sin0-2cos0
sin30-2cos30+zcos0
3sinO+2cos0 8sin45o*5cos45o
- sin3 45"- 2cos2 45o+7cos45o

cos0
,"h+ obx
1

3tan0+2
3tan0-2
-2x [#)'*r,s
l
3x: +2
3 6
-r-o -o

24 cosze + cos4e = 1
cos2g + cos40 = sin2g + cosze
cosae = sin2e............... (i)

tanze + tan4e =4 * #
26. (secA- cosA)z * (cosecA- sinA)2 - (cotA- tanA)2
sin20+ srn.O
.o = sec2A + cos2A - 2secA cosA + cosec2A + sinzA - 2sinA
= 1ffioOl
"*%_ .cosecA- cot2A- tan2A + 2cotA. tanA
sin20+sin2e.co.2e = sinzA + cos2A + sec2A - tanzA + cosec2A- cotzA - 2- 2
+2
co.2o
=l +1+t-2
sin20+(1--cos20)cos20
cos2o 27. :' 7 sinzg + 3cos2 O = 4
4sinzg + 3sin2g + 3cos2 0 = 4

-- 24 --
S.S.C. ADVANCED
 4sin20 +3=4 tan (A + B) = tan135o
=
4sin20 =1 3n
= ... A+B=1Bb'= 4
.81
= srrQ =
lZ = , 35: sin(A+B)=:z
I

0 =30o
+ sin (A + B) = sin30'
tan0=t.r30'= A+B=30o........(il
# e*t.o.(A-g)=r
28. sin21o + sinz5o + sin29o + ................. + sin289o cos (A - B) = cos0o
= sin2lo + sin289o+ sin25o + sin285o + ........ + sin245o A - B = 0" ...........(ii)
{+s{ur (il aqr (iil t
'r
= sin21o + cos2lo + .......... + f+)'
wr) A=300+00 300-00
Z ,B= z
=1*1+....(#)' A15
'B15
_l

11 1
=11 + r=L1-, * tanB
cotA + tanB tanA
36.
cotB+tanA - 1 +tanA
29. cotl8 ' I cot72" co"'ZZ'*----J----------=-] tanB
L tan72"..u.268'-l
= cotl8o I cot72" cos222o * cos72" cos268'] 1+tanA tanB
= cotlSo cot72" lcos222o + cos268'] tanB
= cot18o cot72" lcos222" + sin2221 = t"rrA = cotA tanB
= cot18o tanl8o tanB
-1 37. cose +sinQ -a
30. cotlOo . cot2Oo cot60o cot7Oo cot8Oo
(cos6 + sinO)2 + (cos6 - sino)2
= cotlOo . cot8Oo . cot2Oo . cot7Oo . cot60o
11 = cos20 + sinzg + 2sine . cos0 * coszg + sin20- 2cos0
=1x1r!E'=G . sinO

31. 3[rR sine +cosec0 =2I[6[, sinn0*cosecn1 =2

=1fl=2
+ a2*(coso -sinO)2=2
32. secae -tanae
(cos g
(seczg + tan20 ) (secze - tan2e ) - sin g) = ,[,L- az
38. tan720" - cos270o - sin150o . cos120o
=(sec2g +tan20)xr cos (960o- 909
= tan 2 x 369'+
( 0")- - sin (90" + 60') . cos
=sec2 6+tan26 (go'+ go9
= tanOo - cos9Oo + cos60o . sin3Oo
7
lll
t2
*cosy=2
-0-0*r.r=Z
33. cosx
39. sinQ +cosec0 =2
v+fi{q nrfr qrrq t qe 5:qdrr x= 0o iTstT y = o" td tt
.'. sin0o * sinOo =0 .'. sin2e + cosec'e = (sing * cosece )2 - 2sin0. cosecQ

tanA + tanB =22-2x1_=2

34. tan(A + B) = 40. cos3x = 1
1-tanAtanB
=+ cos3x = cos 0o
11 = 3x=0
.'.x=0
= 41. sin(A + B) . sin (A -B) = sin2A - cos2B
1-( i),[-;)
42. cose + .6 sinQ - 2

-6
5
+ ].o. e . + sino -r tti rtq' iU * +.{i q{l
= 6 --1
=b + .o.f . cosQ +.irf .sinO -1
 = 2 cosec0
=, ."" ('-;) = o' (r)
48. tano t,_.*'5l
e=fr ( r \
= tano t*z,.l
I
43. sine = *,
= tang. cosec2g
'.'o< 0 < z
/ or 1+tan20
I [t_tlf tan'0 I -I
-; z )- L**;h
sin0 49.
Ir+cot2o
0 = 30o, 150o
6 * *+e I qn q*rq tr 1+ tan20
= tan20+1 =tan20
B tanz/lb"- sin260o - |"ot'eo' * * sec24bo tan?a
50. cos (A + B) . cos (A - B) = cos2A - sin2B
51. A+B+C=180"
= 3 x (r)2 -t*l -; ,(J5)' * | ,. (J-r)' + A+B=180'-C
+ tan (A + B) = tan (t80'- C)
331
=$-Z-r+ rx2 +
tanA + tanB
1- tanAtanB = -
tanc
12-B - 6 +1 tanA + tahB = - tanC + tanA tanB tanC
= =7 =
4 + tanA + tanB + tanC = tanA. tanB . tanC

- lnl lnl 52. ,ficosQ +sin6 =1

45. coszlZl -.o.'IoJ
+ Jd1 +, sin0
1
- , Itri mr iU * +.d vtl
,cose
1
+ cosSOocose +sin3Oo sinQ- ,
19199 + cos (30'- e ) = cos 60o
=--
216161616 + cos[-(g -30")] =cos6oo
46. secA + tanA =4
+ t(g - 3o')l = 60"
1 sinA
< cosA? cosA: =+
1+sinA=4cosA
o=90"= I
= 53. cos S + (-1) sin 0
qH r{s'e,i q-ti *
i--

=+ L+2 sinA+sin2A= qFI = J(r)2*(-t)2

16cos2A oTtq-frTq =.,|t*t:Ji
+ sin2A+ 2sinA+ 1= 16 (1 -sinzA) 54. sin105o sin7So = sin (180" - 75") sin75o
17 sinzA + 2sinA -15 = 0 = sin7So . sin75o
=
+ 17sin2A + l7sinA - l5sinA - 15 = 0 = sin [(go'+ 45))2
+ (sinA + 1) (17 sinA - 15) = o = [sin30" . cos45o* cos3Oo sin45"]
sinA+1* 0
15 =(L,.a*€ +)'
sinA =
-
LI [2 Jz 2 Jz)
tan0 r_ tan0
47.
secO-1 secO+l
Isec0+l+sece-ll
tanQ
L--Grr_, l
tan0.2secO 1 cos0
tan2o - z ' .".e .*e bb.
 1l .\ 7[
(r+sine)2
- (t*sino)2
dffr-{ tano -131R tan3 = J3 :- (r*sino)2
=[.o.e -
) -;7t--(r*sino)(1-sino)
t^nf, < tang . r"rr{ _l-l
/t+sine\
- [t- sinoJ
lq' e '1 ( r 1 \f 1 1\
l-r- ll---
61. ' I

tan50o. tan4Oo [cos0 cot0./ [cos0 cot})

= tan (90'- 40") tan40' = (sec 6+ tan 0) (sece- tane)
= cot4Oo tan4Oo
-1 = secz6- tan2g
_I
57. sin6 cr + cos6 6x + 3sin2 cr . cos2 cr -1

= (sin2 c, )B + (cos2 cr )3 + Ssinz cr . cos2 c, sin

_r- 0 cos0
= (sinzcr * cos2o)3 - 3sinzo . cos2o (sin2o * cos2c,) + cosO cos0 - o
62. sinO cos0 -4
3sin2 o co.2 .t,
g o cos0 cos0
= (t)3 - 3sin2 cos2 61 + 3sin2 . cos cr

-1 tanO +1 5
+-=-
58. sin6e + cos60 - tan0-1 4
= (sin2 o )3 + (cosz e )3 + 5 tane- 5 = 4tane+ 4
= (sin26 + cor20 ) [(sin20 )2 + (cos26)2 - sin2e cos20 ]
+ tan0-9
= [sinag + cosae - sinze . cos2e]
92 +1 82 4l
= [ (sin2g + cos2e )2 - 2sin20 . cos20 - sinze . cosze ] ' tan20+1 =
tan,eL g'-L= 40= 40
= [1 - Bsin2o. cosze] sin0- cosO= 0 + sirie= cose
.'. 0 = 450
= [, - 9(z.irre..o.e)21I
14 sin (nl2- e ) * cos (nt2- g)
'.'0<sin220<1 11
= cos6+ sino= cos45o * sin
OS
33
450 =
a*o= J2

+sinz2eS A 64. sece+ tan0- P

33 +(secg+tang)2=P2
0Z -4sin22eZ -A
( t sino\'
33
1>1-Osinz2eZ1-A = [.o.e *.o*J =
'=
(t + sine\'
3l =[.".e )="
1r1-Asinz2ez 4

...3TFrfitrq rTl;r - 1 ((r + sine)')

59. ct + 0=90o'.' a=29 = [ l-.ir,e )="
+2p*F=90' +3P=90o _ 1+ sinO _-' ,,
P =30' ' l-sino
a =2F = 60" =sin0*1=P2-P2sine
cos2cr + sin2P = cos260o + sin230o +sine+P2sing=P2-L
(t\2 (r\2 1 1 . (P'-1')
=[zJ.[;.,l =z*z=,
1
.'.srn0=
u+PrJ
cot30o-cot75o
65.
(tans iseco)2= fg4*+)'
Icos0 cosOJ
tan(90" -75") - tan(90o -30o )

cot30o -cot75o
cot75o -cot3Oo
 a
l.
-- 1
66. cote.cote - cosece.cosec0+ (sin2250*cos2250) + S(tan 71. cose*.*,e=2

50. cot 50 . tan 150. cot15o. r= )

1
= cos2g- 2cosg+ 1=0
v3' + (cos 0 -1)2 = 0
1 .'.coso = 1

= cotzg - cosecze + 1 + r/3 "u6- 1

cos6e* sec6e= (1)6 * ,5= 1 * 1
-- 1+ 1+ 1
-1 -2
67. cos 0.cosec 23o =t 72. ZC = 1900 - (0oo + 300)
= 900
+cose= .*#tr= sin23o ... 1g0o = n"

+ = cos (90'
cos 0 - Z3o) fi"
... 90"=
cos0= cos 670 -x90
=
.'. 0 = 670
_ Tc'

68. (sin250+ sin2850) + (sinz 100 + sin2800) + ..........+ (sin'400 + 2

sin2Soo) + (sin245o + sin290o)
= (sin250 + cos250) + (sin2100+cos2100) + ............+ 5.
(sin2400+cos240o) + sin2450 * sin2900
73. ;;T. ,.""2p, + 3 sin20

= 1'* 1 * ........upto 8 times +

/r\' + t'z = 5 cos2e+
2
+ 3 sin20
[1q.J "o.".re
= 5 coszg* 2 sin2e + 3 sin2g
1
=8* r*l = 5 (cos20 + sin'e)
-1)
1
74. xsing=ycos0
=Q- "2
(n 0\
6e. tan
lr- r)= !6
0
=+ cot
2 = cot 300 eF[, a ABC d,
e P=y
= t= 3oo bx
e= 600
...h=r[r, +y,
".
1
cos 600 = 2^ 3lt[, x sina 0 + ycoss 0 = sinO cosO

2sin0 - cos0
7n. --------:------:
cosu+slnu =
r ,,r x )'= t
=.( --r-l'-r[5)=ffi
=15J
2sin0 cos0
sin0 sin0
-r
cos0 sin 0
sin0 sin0
-+_
2- cotO *2 *u2
)-*=1+ ,[r\y'=t
1
-
- cot0+1
lx'+Y'
+2-cote-cote+1
-= f+rz-,
cotg=
=2 1
75. (1 +tan 0 + sec 0 ) (t * cot 0 -cosec 0)
1
=(r*sino *-l-)[r*
cot9= coso
, -r)
\ cos0 cos0/( sin0 sin0/
 (cos0 + sin 0 + 1 \(sin0 + cos-1) 2 2 cos 0 + 2sin 0 - 2sin 0.cos0
-
I cos0 i I sinO ) 2 cos 0(1 - cos 0)

_ (cos0+sin0)2 -$)2 2(1- cos0) + 2sin0(1- cosO)

cos 0.sin 0 2cos0(1 - cos0)

e * sin2 e + 2sin o.cos o -1 2(1 + sin 0)(1 - cos 0)

"or2
cos 0.sin 0 2cos0(1 - cos0)

1+2sin0.cos0-1
_ 1+ sin0
cos 0.sin 0 cos0

_ 2sin0.cos0 o .oa4 o sin4 cr

--
cos 0.sin 0 ' ,ir2p '
-i-
-<

sin2 cr = cos3 c[ (il "o.2p

qR cr =p td t a v+*,r', 1tr *en tr
= sinz cr = cosz o.cos c[
sin2 cr . .ora B _, sin4 B _.,
:) , =coscl, O

cos'G sin'ct
I
^ -I

cos- cr,

1 qeT = 86o0 t+ug, qeT = 69 fraE

=> tan2 G = cos 0, 1

51
cotz cr
1
....(ii) . 63+14+
cos c[ 60 3600
ffi, 226800+840+b1
cot6 cr - cotz cr 3600
= (cot2 o)3 - cotz a 22769L
11 - 3600
=;J;-."r" (Hft. (iil sl ffi1 6$ q1;
75897
- 1200
- 1 cos2 cr
-l;5;
sin2 cr
elG[, lc =
frtm-
-;;5;
75897 Tc /zgrr \"
aor3o
1200 180=lJ7[
(8000 /
I

=#=1({Fri.
cos'ct
(il rt)
80. sec0+tan0=.,8

77.
sin0-cosO+1 (sec 0 + tan e), = (Jd)2 (+d d{s' erf q,ri qr)
sin0+cos0-1

- cos,)
sin 0 + (1
- sin0-(1-cos0) =(r*tino)2 =B
'#a .tffi [Rationarising qri q<] \ cos0/
(1+sin0)2 o
_ l,r1q * (1 - cos 0)2 + 2 sin 0(1 - cos 0) - o -u
- cos'0

(1 +sin0)2 o
_ sin2 0 +1 +.or2 e - 2cos0 + 2sin0(1 - cos0) - 1-sin"0 o
-U

sin"0-6@ 1+sin0
-----B
+ sin2 0 + cos2 0
- 1-sin0
1
- 2cos0.sin 0+ 2sin0 - 2cos 0
sin2o-t+2coso-c;Je +1+sin0-3-3sin0
1+1 + 2sin 0 - 2cos0.sin 0 - 2cos0 +4sin0=2
-
sin20 = 1-cos20]
.or2e-.or2 e
2cos0-1+1-cosze- L

.'.slnU=-
1

2
2+ 2sn0- 2 cos0 - 2sin 0.cos 0
8L.2 sino+15cos2 a=7
2cos0-2cos20
 + 2 sin cr + 15 (1 - sin' a) = 7
+
=+ 2 sin 61 15 - 15 sinz
q= 7 --(L)' +(Bsincr)2
+15sinzcr-2sino-8=0 \srncr/
+ 15 sinz ct- 12 sin 61 + 10 sin cr- 8 = 0
+ 3 sin cr(5 sin a - D+ 2 (sin cr - 4) = 0 =( -2-- gri, o)' * 2, -2-x 3sin cr
+ (3sin a + 2) ( 5sin cr- 4) = 0 \srncr / stncr

42 t... a2 + b2 = (a _ 5)z + 2abJ

sincr= bqT -E
/ d t2
[2-Ssin"a -- II
ABC {, -l
-l
- -----
I
r'tO
I

^ I srncr )
nc:{s;'-1+f =a
-[z-g(r-"or2o)] *rz
.'. aot c, :I I srno )
4
(a2
- b2) sin 0 + 2abcos 0 = +b2 (z-s+g"or2.,,)t2
fr rrq. +bzrt qrtr ti qt,
a2
=l-------:-l
srno +n
az I )
(a2 -b2\ ^ 2ab ^ a2 +b2
-[Bcoszcr-r))'*rz
I sincr )
=\^:-??!sino+
(ao +bo)
. ?\ u-
(ao +b") "oso
=1 ......(i)

qHrf6 lR= a
(PR)'z=(PQ)'+(QR)'z
(a2 + b2)2 = (zab), + (az _ 5z;z 2t
dt relation II-FI *t
-b2
a a-b
, -rR
cos ct, =;r+;,,
u2
srn ct, -7;P
2ab .,
sut- ct
+12

€++,.{q (il t,
cos o. sincr + sino. coscr =
sin (0 + a) = sin 90o
1
srfrm t oTrrn
-('(=*)-')' sr qliIII rTH 0 fi
sin2
cr
0+cr=90o
tan 0 = tan (90"
ti t d gtt vrfrfi{ur Hr qfilq rtFT 12 drlTt
- cr) = cot cr
82. qHrF6 AB = G + z) rtfr.
=u'.:,o' =11ur-5r; .'.BC-rtfr.
2ab 2'o - g;[G[, A
{F+ fdq exercise t5T 75 to|-{ qr"rtion idl

t\
eJilz-f+G+2)z
- sin2A 1+cosA sinA + 2O=**x2*4+4x ,\'*
84.
1+ cosA sinA 1- cosA
+ 2f+4x-L6=0
_ 1 (1 - cos2 A) (1 + cos A)(1 A) - sin2 A
- cos
= f+2x-8=o
1+cosA sin A(1- cosA) +x'+4x-2x-8=0 82C
.,.x = 2
(1; cos2 A) - sin2 A W,
=1-(1-cos
sinA(1- cosA) Cos2A - Cosz C
=l-1*cosA+0 ( (
+ \2 z )2
=cosA
85. 4 cosec2 cr + 9 sinz cr
-t6j (.61
-t_t_t_t

= l,y --:-+
1
9sin2 cr 164
sln- c[ 20 20
 =y=3
205 slr[,
88. A iTsr B qm otq tr
A+B=90o 5
A=90o-B -g=5
W, 77
3
SinA CosB + CosB SinA- tan A tan B + Sec2A - Cotz B
= sinzA + cos2A - 1 + 1 * tan2A- tan2A
[...sec'A=1+tan2A]
=l-1+1
-1
89. 2- cosz 0 = 3sin0. cos0
ti mm cosz o t qrtr ti w,
2 -cos2 0 3sin0.cos0
.osZ
6-

e .or2 e I- 2I
cos0l --------=- l= 4
= Lcos'01
2 3sin0
- aor2e ^- coso +-- 2 4
=) 2secz0-f=3tan0 cos 0

+ 2 (l * tan2 0) - t = B tan 0 1
cosU=-
=2+2tanz0-f=3tan0 2
+1+2tanz0=3tan0 0=60'
=2tanze-Btan0+1=0 93. cos 0 + sec 0 .... (i)
+ 2 tanz 0 - Ztan 0 - tan 0 + 1 = 0 = JE
+ 2 tanO (tan e - 1) - 1 (tan 0 - 1) = 0 cos80+sec30
=+
(2 tan e- 1) (tan 0 - 1) = 0 = (cos 0 + sec 0)t - 3 cos0.sec0 (cosO + secO)
.'.tan0=19I 12 = (JE)3-Bxcoso* -LrJIl
90. AB = 2.5 tfr. = BJ5-BJ5=o
51 95. qR sin 0 * cosec 0 =2, iltl
CosB = 0.5 =
102 sin"0*cosec"0=2
sin70*cosec70=2
CosB =;1
n n 3n lrc qfi
CosB = Cos 60o
96. ,* g * tanf tan-= -ra';
12
B=60o
-tann tan
n
t*r(Y-I),un[l- r )-sir,21
1 8L2\28)[2L2)6
Cos6o. = 2
tr tr
cotlcota
8 L2 8 t2-(f)'
= tan tan
Sin6o. =+=ffi 12)
fi tr 1

J5=,q.c 8 8 L2cot L24

=tanficotfttan

2 2.5 _13
-I
44
=+AC_2bxG=sJ5
70x2 4 97. cos (zg'bg'28'') + tan (10o6'82")

91. AB= JW = cos (g0" - 10o6'82") . #ffiffi

-D
-U
= sinlOo6'32"+!
Jt-u2
 d

.'.tane= {r_*
lt-r
uJL-u2 * u
102. sec0+cose,-2
=9
Jt-J
+2(sec0+cos0)=5
---G- a(1+ {1- a")
^
+ z( I
[cosO
+"o.0) = b
)
tan 9o.tan27'.tan(90o - 27").tan(90'- 9")
= tan g'.tan(gO" - 9').tan27".tan(90' - 27) _ *.o'2 e)_
+ ,[r
"[ coso u"
= tan 9".cot9o.tan27".cot2'lo )-
=1x1=1
99. Erd'r qme t fo =2(t*cosz0)=5cos0
+2+2cos20=5cos0
sec20-tanz0=1
(secO + tanO) (secO
+2cos20-Scos0 +2=O
- tanO) = 1
0 -cos 0 + 2 = 0
5 = 2cos2 0 - +cos
g
;
(sec0
- tan0) = 1 + 2cos 0 (cos
-Z>- 1 (cos 0 - 2) = 0
+ (2cos e - 1) (cos 0 -2) = o
.'. sec0 - turr0 =
2 '2cos0-r=0{[ cos0-2=0
b
2cos0=1{I cos0= 2
.o.2 e1.or".2e* 1;
100. /cos20+msin20= 1
.or""2e -1 cos0=rq 2

.or'e[-1- *r
Hf*,r o < e. gur tr
=/cos20+msin20 -
I sin" 0 ;fr*
cot2 0

.'. cos 0 = I
e [r +.i.r2 e ) 2
+ /cos2 0 + m sin2 0 =;pr[-;;5-.J
"o.2

+ /cos2 o + m (1 - cos2 o) =#'$o+sin2e)

+ - m cos2 0 = 1 (t + sin2 0)
/cosz 0 + m
+/cosz 0 + m-m cos2 0 = 1 + 1 - cos2 0
0)
= /cos2 0 + m - m cos2 0 = 2- cosz 0
(sec0+ tanO)(sec0 + tan 209
,".2 e- an2 o 79
= /cosz 0 + m cosz 0 + cos20 = 2 - m
+cos2e(/-m+1)=2-rr^
=
(sec 0 + tan $'= 2P
79
2-m
- {-m+1
.'.cos20= .
( t sino)2 zog
-
1.-=-I -m+1 \cos0 cosO/ 79
-r--r-r:-
' sec;v=
- cos'0 2-m (t+sino)2 zog
3;rG[,
I cosO ) 79
tanz0-1-secz0
1+2sin0+sin20 2og
-t
- l. -m+I e 79
2-m "or2
= 79 (1 + 2sin 0 + sin2 0) = 209 (t - sin, 0)
2-m+ (. +m-L
= 79 + 158 sin 0 + 79 sinz 0 = 209 - 209 sin2 0
2-m =+ 288 sinz 0 + 158 sin 0 - 130 = 0

_ (,-l -158r@
2-m sin0=
2x288
 -1b8 tJr4s64+L4g760 BC = Gqt_(ABF
y2 -12
v
576

-158 + 418 3;rG[,

576
lglc = 90" - 17" = 78"
-158-418 -158+418
576 ' 576
sin73'=#:ry
-576 260
576 '576 slrl,

65
Sec 17" - sin 53o
=-'l -'144
tan 57o + cot 37o
104.
tan 33o + cot 53o

sin 57o+-sin 53'

- cos57o cos53"
cos57o cos53"
+_ 2
sin 57" sin 53' x

Y('lY' - *2)
sin 57o.cos 53o +cos 57o.sin 53o
cos57".cos53" 107. tan 20 tan 30 = 1
cos 57o.sin 53'+sin 57".cos 53o 20+30=e0'(Rqt EsTr +)
sin57".sin 53'
50=90'
_ (sin 57".cos 53o +sin 53'.cos 53' ) x sin 57o.sin 53o .'.0 = 18o
(cos 57o. cos 53' + sin 5 7o. cos 53o ) x cos 57o.cos 53" 3;lt[,

= tan 57o.tan 53o 5o

= tan 57". tan (90" 37')
2cos2 -r
- 2
= tan 57".cot 3'1"
5 x18
^./nx\ q \ =2cos2 -,
1oE. ,._[rj _,- *?
2

=2 cosz 45" - L

"(.""', )* -' - 7
1
-=z("in!)r:"* =2,( L\' -,
\Jz )
+2= *'*l,2 =2r!-1
2
=0

['.', = 1, sin 108. Ze,+ ZB - eo' (tiqr EsTr +)

t=r, Sec2 A + Sec2 B - Sec2 A. Sec2 B
= Sec2 A + cosecz A - Sec2 A. cosecz A
)2-( ,-L\' *, 111T -------?-
Ir/
* b2 = (a _ b)2 + 2ab7 A A .o.2 A.sin2A
l.: a2 cosz sin2

( sin2A+ cos2A -1
x-:r\2| -0
=l Ir/
= sinzA.cos2A
1-1
. (.-1)=o =---------a----- -:0
|t.,/ Sin"A.cos'A

vrdtq ABC d,
tano 1- tano)
106. A 109. (2Cos20-r',[1+ *
[1-tanO 1+tan0/
 ellltr 170" 100
(r ttun- t"nelz ) L-
22
t/
.Y- -
= 12cos20 - tan o) )
"[-1r-
'r[ tunexl+
x=85o,y=5"
-

114. tan0 + cotO = 2

1 + tan2 0+ 2tan0 + 1 + tan2 0- 2tan0
+ I
= cos 20
1- tan2 0 tan0 =2
tan0
_ co20x 12+2tan2 0) + tan20+1=2tan0
-I--. ,rn2 e + tan20-2tan0+1=0
cos" 0 (tan0-1)2=0
tan0 = 1
_ cos 20 x 2(t + tanz 0) cotO = 1

1cos2 o - sinzo tanlooe + cotlooo

Cos2o - (1)100 + (1)100
cos20x2Sec20
=1*l=2
- 115. qFIT t*. {qf6elg H-r r+d' +tot e tr
cos20 x Sec20 ^
5n
t... Cos2 0 - Sin20 = Cos2 0l o+o+ =n
_Z 9
-6

1-l0. 2 (cosz 0 - sin2 0) = 1

2g*5n =n
1 9
=
(Cos20
- Sin'O) = z
20=9
! I
+ Cos20t-=
2
+ Cos2 0 = Cos 60" e=49
=20=60o
.'.0 = 30o 116. sec0 = x+: 1

rTr'tT ffi { qtq a aqr u tr n,

111.
a+b=135o h 4xz +l
fi
= 135 " 1g0"

3n
a*b= (i)
4

ll
a-b= (ii)
, = rf Qx' -LY
3n TE. -+ -t
Tgr dur =a= =T*, secO * tanO
2
+L Axz -l
3n+2n 5n =-4x24x 4x
-_-=-
4x2 8
113. Cos x = sin y =-8xz
sin (90'- x) = siny 4x
90'- x=y =2x
x+Y =90o "" (il
cotG-40)=tan(so"-y) sec0ltane _ 5
tt7.
+ tan [90'- x + 4o) = tan (so'- y) - tan? 3
sec0
...90'-x+'40o=50'-y +3(sece +tar,l)=5 -tan0) (seca
+ 130' - x -r 50'- Y +3sec0 +3tanfl =5secg - 5 tanQ
x-y=800....(iil
+2sec0 =8tan2
*--90o+80o 90'-80" ,Y=- Z-
,
 1 4sin0
cosd=-cosd

.'sine=i
1
. =_x7
180
22
118. rTf;TI acosO
- bsin0 = r 630'
(acosO-bsin0)2=x2 = = 57o L6' 22 " ( c4q$rq )
?
a2cos2e + b2sin20 - 2ab sino . coso = x2 (i) 123. sin0 r cosecO = 2
(asin0+bcosO)=c
(asin0+bcos0)2=c2 sin0+ | =2
sin0
+ a2sin20 + b2cos20 + 2absino.coso = c2
............ (ii)
+ sin2o*1=2sino
(il + (ii) + sin2e-2sino+1=o
+ b2sin20 - 2absino.coso + a2sin20 + b2cos20 +
a2cos2e + (sinO - 1)2 = 0
2absinO.cosO=f+cz sinO = 1

+ a2cos2e + b2cos20 + b2sin20 + a2sin20 = *2 + c2 cosecO =1

a2cos2e + a2sin20 + b2cos20 + b2sin20 = *2 + c2 sine0 + cosece0
=
+ a2+b2=*+c2 = (t)e + (t)e = 2
+ ,2=a2+b2-c2 124. sec0+tanO=2+ JB
x= + Ju2 *b2 -"2
cosO+cos20=1 (i) ^ (z* .6.)' * r
secO =
cosO=1-cos20 2(2 + li)
cos0 = sin2e
sin12o + Bsinloo + Bsinso + sin6o - lfr secg + tang = a-l
1
llI a2+l
= (sin20)6 + 3(sin20)5 + 3(sin2e)5 + 3(sin20)a + (sin2e)3 -f l.'.sece-- - I

L2a-l
|

= cos6e + Bcos5o + Bcosao + cos30 -f= (cos2e + cos0)2 -1

=(t)2-t (IIfr. (il t)
++1+ !^Ji=+t
=Q = seco -
4+2J5
l2O. x =asecOcosO

t""o"otO (i) _10+4€

i = ++zJb
y = bsecO sin0

Ib = ,"" osin$ (ii)

z = ctan0

7:tan} ................ (iii)

c

x' .y' r.o

z'
o o
a' b' c'
= (sec0 cosQ)2 + (sec0 sing)2
- (tan0)2
+ sec2$ sin24 tan2o
p =./(Jr)'-(1)' = J+ =2
= sec2e cos26 -
+ sin2$) - tan2o
=
=
sec2e(cos2o
sec2e - tan2o = 1 -
sino+coso= +*+ ='
G.6.'6
121. coslo cos2o cos3o . cos177" cosL78o cos179o 125. sin2250 + sin2650
= (coslo cos2o cosSo cos177o cos178o cos179") x
cos90o = sin22bo + sinz(9oo - 25)
= (coslo cos2o cosSo cos177o cos178o cos179o) x 0 = sin22bo + cos225o = 1
-0 sin0 cos0
L22. nc = 180o 126. xy :-
 sin0 x (sin0 + cos0)' -1'
cosO y sin0.cos0

sin2 0 + cos' 0 + 2sin 0cos0 -1

tan0 =I sin0.cos 0
v
2sin0.cos0
_-=.,
sin0.cos0

131. sin(A - e) = !
2
Y sin (A - B) = sin3Oo
sing-cosQ = -L- A-B=30" (il
J.\v' E.v'
cos(A+B)=1
2
cos (A + B) = cos60o
A+B=60o (ii)
127. sec0 * tanO = !6 (il aqr (iil t,

o_30+60:90=45o
sec,=Etr
- 2$
uvvv t:::::?1';',]
Lsecu: 2^ ]
22
g:60-30=30=15o
3+1 22
Sec0:
-.-
2J3 ao- 15"xr - t
,LU
180 rz
A 132. (1 + (r + sinB) (t + siny)
sincr,)
ssg Q = __=
2J3 = ( 1- sind (r - sinB) (r - siny) = x IT;II
ffi, (t + sincr) (r + sinp) (t - siny) = .r ..............(i)
iIsTI (t - sincr) (r - sinp) (r - siny) = x ..............(ii)
secU
^2
-: (il x (ii)
Js
(1 sin2o) (r sin2B) (t sin2y) = f
secO = sec3Oo
- - -
cos2cr cos2B cos2y -*
x = + coso cosp cosy
...0=B0o =tan3Oo= #
1S3. fr asin0+bcosOd d
128. tanO * cot0 = 2
tan0=cot0=1 erttrorq qn = J;\b,
... tannO * cotn0 . sTTif,fiirTT TTFT = J2\ 3,
=1'+ 1'
=1tL=2 = J4+g
129. sin(60'- e) = cos(V - 30') : JtE
sin(60'- e) = sin (go'- y + 3o')
t34. 152 (sin30" + 2cos245' + 3sin30o + 4cos245o + ........ + 17
= 60o-0=90o-V+30' +
=) 60"-0+v-30'=90o sin3Oo 18cos245')
y_0=60o = 152(sin30'( 1 + 3 + 5 + .... + 17)+ cos245o(2 + 4+ ....+
tan (y - 0) = tan60o 18)
:6 =r52,.|tr * B + b+......+rr,.[#)' (r* n+......+18)
130. (1 * cotO - cosec0) (1 + tane + sec0)
sino
=( r*coso
( sinO---1-')[r*
sin 0/\ cos 0*,cos 0/)
(sin 0 + cosO - 1 \(cos0 + sinO + 1\
- sino coso
[ /[ )
 -76x9x19
=4x19xgx19 = 6x2+ +r?
8
=(2x 19 x 3)2
=Wrfo. *-r Wi qri =L2+1=13
140. 3(sec2e + tan20) = 5
l3b. 1-.nu +--1 . +Bsin2a \R
+ tan" cr 1+ cot'cr
+ sec'e+tan2o=;
41 + __: + Bsinz cr
"
sec- ct cos ec'cr, :9
+ 1+ tan2 0 + tan2 0
3
= + lsin2cr + 3sin2cr
4cos2cr
=4cos2o+4sin2cr + L+2tanze::3
=4(cos2o+sin2o)
-4
1 3 0-
+ 2tan2 g =?
3
136' +zsrn'o
r..otT* r**rr=
tan2e=!
130+ --------- + 2srn- 0 = 3
cosec"O sec'0
= sin2o + Bcos2o + 2sin2o + tan2 o =L -.6
= 3sin20 + 3cos20 + tanO = tan3Oo
= 3(sin2o + cos2o) 0=30"
_D
-r)
137. sin0 r cosec0 = 2
cos2o = =l
sin0=cosecO=1 "o.60"
sinse + cosec5o 1
sin30" =
= (t)b + (r)5 2
-2 cos20 = sin0

.".e[l1'-9+-g!L 11
188.
\ cosO 1+ sin0/)-zt"rr,e
1r.1
cosecO-cot0 sinO
+ sin_o)'+ cos'01_zt^n,
_ .".r[(t e cosec'0-cot2 0
I cos 0(1 + sin 0) -] -.i"e
1

cosecO-cotO
* sin2 0 + 2sin0 + cos' 0
- --- ^[r
66^ l-l I
2t".,'e (cos ecO + cot O)(cos ecO
- cot 0)
L cos0(1+ sinO) ]- (cosecO
-cot0)
- cosecO

_ seco(2 + 2sin9) _ 2tanz o - cosec0 + cot0 - cosec0

cos0(1+ sin0) = cotO

2sec0
o,
142. cosO + sinO - J2 cos0 (i)
=--z(,A7;zO
cos cos0-sin0=a (ii)
0
(i)2 + (ii)2
= 2sec20 - 2tan20
-2 (cos0 + sin0)2 + (cosO
- sin0)2 = ( J-Zcoso)2 + a2
139. S(sin-r-cosr)a + 6(sinx+cosx)2 + 4 (sin6x+ cos6x)
rrll 45'{Gfi q{,
t5t - t'']
x
= 2(cos20 + sin20) = zcos2., + a2
= 3(sin45o - cos45o)a + 6(sin45o * cos45')2 + 4(sin645" + [:';3.;3
cos64bo) + 2 2cos20=a2
= 2(sin20) = a2
o. u(# . -{(#)' .(#)'} a .[ sin0
=
i)'.
143. tano = 9
4
=.'(#)'..{;.*}
 p=3 (arr)' 180 Bn
b4 l-
\5/
l=-
Tc -5
6=,[g44'=5 = 108o
148. tanO * cot0 = 2
h5 tan0=cot0=1
cosecu -
pB tan20 + cot20
11 =1+L=2
144.
1. d, e - 1*.orr 149. 3cos80o coseclOo * 2cos59o.cosec31o
' = 3cos80o . sec8Oo * 2cos59o . sec59o
11 f-
=3*2=5
sec'o cosec'o
cos'A(sinA + cosA)
= cos2e + sin2o = 1 150. cosec'A(sinA cosA)
145. tana = ntanF
t -
(sec2A
tano - cosec2A)
ir=- (i)
tanB = cos2 Asinz oI sinA + cosA) * .i.r, A cos, oIsinA - cosA )
sino = m sinB \sinA-cosA/ \sinA+cosA/
(seczA
sincr - cosec2A)
Iii: (ii)
--srnp ,
^, ^lsinA+cosA sinA-cosAl
=cos-Asln-Ar--!- |

[sinA-cosA sinA+cosA]
m' sin2 o tan2 p
=;t"r_B ^. (sec2A - cosec2A)
* tur? o
-l
+ cos4)' + (silA -cosA)'
m' sin'u . sin'B x cos2 cr -- o[(sinA
- Asin2
= cos2
= n' - sin'P " .o.'B x sin2 cr L sin2 A - cos'A l
m'
(t 1\
cos'cr
---) n' (cos'A sin'A,/
-= cos'p
+ m2 cos2B = n2cos2ct 2 tr _:_2 [2(sin'A+cos2A)
= cosz Asin'A ^
= m2(1 - sin2B) = n2cos2o sin2 A - cos'A
+ m2 - sin2cr = n2cos2c[
:+ m2 = n2cos2cr * sin2o : 2(sin'A-cos'A) _.,
,,
m2 = n2cos2o + 1- cos2o
GI;;r\ -.o.%) =
= 151. xcosO - ysin0 = I (i)
= m2- L = (n2 - 1)cos2cr
xsinO*ycosO=4 (ii)
. , m2 -1
cos-c[=-
(il2 + (iil2
n" -1 Gcos0 - ysin0)2 + Gsin0 + ycos0)2 = 22 + 42
+ *coszg + y2sin20 - 2xysino . cos0 + lsin2O + y2cos20
146. cosao - .inng =3 + 2rysin0.cos0 = 20
= x2(cos2e + sin20) + y2(sin20 + cos20) = 20
+ (cos2e + sin2o) (cos2e
- sin2o) = ? + x2+y2-20
3
152. sin20-3sin0 +2=0
+ 1(1 . sin2o - rirr'g) = + sin2o- sino - 2sino + 2= O
3 + sin0 (sin0 - 1) - 2 (sinO - 1) = 0
(sin0
= - 2) (sinO - 1) = 0
+ 1(1 - sin2o - .lrr'g) =
3 :+ (sinO
- 1) = 0 {I (sin0 - 2) = O
sin0=1 qTsin0=2
l-2sinze =?3 0 = 90o (sinO d' ffi angle tfi"T q;T 2 Tfr Etdf +)
153. cosax * cos2x = 1

r47 r'=[lao)" + cos4r-1-cos2,

In,/ + cos4, = sin2x
tan4x + tan2x
 :-l-
sina x sin'x
cos4 x cos2 x
x
sina t_
cos4 x
x
sinz cos2 x
= sin2x * cos2x =1
154. 7sin20 + 3cos20 =4
+ 4sin2o + 3sin2o + 3cos2o = 4
+ 4sin20 + 3(sin20 + cos20) = 4 [uR .o.". 0 -cotl= a d coseca -a'+L **,-l
+ 4sin20+3=4 lzul
158. x sin45o = y cosec30o
+ 4sin20 = 1
x cosecSOo
t-
. o^ 1
srn'0 =
= Z ri,r+S"

x2
sinO = tl2 t= I
0<e<90' J'
sino = 1 L:zJi
2 v
sinO = sin3Oo
0= 30o. *r
=[;J
-( "\^ =\ztz)
-(,)^t;\n =64 = 4s
J5 ;"
cos3Oo = cos3Oo = 159. rsin0=1 (i)
2
IbD. 4tan20+ 9cot20 r cos0 = !6 (ii)
3ft156q qra =2J4* = 2 x 2 x B = t2
(i) * (ii)
1
['.'atan'o+bcotze 6-r erferfiflq qH] tanU =-----:
t_l JA
l=zJaa ]
rEtanO + t
1 'l
186. sin0 - cosO = 7 =JBxf+l =2
JE
sinO+cosO=x
(i)2 + (iil2 sinA sinA
160' 1+cosA-1-cosA
sin20 + cos20 - 2sin0 . cosO + sin20 + cos20 + 2sin0 . cos0
1, ,
_--'- , ^ [(1-cosA)+(t+cosA)
=.i.ra[
4 1-cos2A

-3 2=1+
1

xz
sinA x2
4 sin'A

- sin2A = 2cosecA
1
-3 2-1= xz
4
161. Same as 142
1= *'
2
162. qqsrE5q Hr r+d. eiil: tfitur _(8-2)x180.
v,t; 8

2 = 135o

n
157. cosecO -cot|=12 =135 *
190'
3n
4
{ * *&ry$!e!e!ala!dde!!a!iarsi:!r*n*rlcr*i*ryii}il*n(@.5!B.+1i:r!;i!firt::i&, j;, jri I ,r ! irEiaidffisffi s $&

cot30'= rE
r63. sin0=P .................. (i)
(sin223" + sin267')
h 167. 2cosec2}B" . cot267" - - cot267"
.o.e:|h .. (ii) -=
2 cosl67'-l-cotz67o
r vvu v '
airrr 28.' ,ior 6Z"
(il x (ii)
sinl 23'
sing . coso =+rP
.
sin'23o sin'67o -L- cotz 67o
hh
2
sin0.cos0: =;;ftF -r-cotz 67"
= +!
-- 2cosec267" - L - cot267o

= sino.coso = 1g
2h"
tqgq tl = cosec267o + cosec267" - L - cot267"
= cosec267o - 1 + cosec267o - cot267"
2sin0 . cos0 = 1
= sin20 = 1 =cosec267o-1+1
= sin20 = 90o = cosec267o
= 20=90o = sec2z}"
= 0=45o
164.5 tan0=4 16g. cos43o=
4
tan0 = U

5sin0-3cos0,
5sinO + 2cos0
*'+y'
5sin0 - 3cos0 p=y
cos 0
5sin0+2cos0 sin47o=cos4Bo=
cos0 ffi
5sin0 3cos0
sin 43o:
cos0 cos0
cos 47o = --J-
5sin0+_
2cos0
cos0 cos0 x
cosfl"- y
tani;lo=sin47]
5tan0-3
169. Same as 144
5tan0+2
_4 sec0 + tan0 Z+ J3
5x ,u'
ttn

sec0-tan0 - 2-Ji
5
-3
_4
5x: +2 1 sinO
5
cosg cosg _ 2+VB
4-3 1 = 1 sine-z-J5
4+2 6 cos0 cos0

165. Question rffA tf 1+ sin0

166. 2 (cos2e - sin20) = 1 .r.e _z
-
+ Js
+ 1-sinO 2-Jg
cos'o-sin'o=! cos 0
2
1+sin0 Z* Ji
cos2o = 1
2 = 1-r*e -r-G
cos20 = cos60o
= = (z-J5)(t+sino) =(r*.6)(t-sino)
20 = 60"
0=30o + 2- J5+2sin0-.6sin0 :2+J5-2sin0-..6sine
 = 2sin0-JdsinO+2sin0+rEsin0 =zJB :22L
2
= 4sin0 =2J3
174. cos20 + sec20 H'r qftrrl rIIT =2J1"7 =2
= ti'e= 6 175. cos0 * sec0 = 2
2
sinO = sin60o |
= + cosO+
cos0
,=Z
fi
e=60.= E + cos2e*1=2cos0
:) cos2e-2cos0+1=0
]l71.. sec0-.o.e=9
2
+ (cosO
- 1)2 = 0
cosO=1
1 ^3 =
---\ -COSU=- cos0 = cos0o
cos0 2 0=0
Scoso cos100 + sec110
= 1-cos2r- 2 = 1+L=2
176. sec0 - cosecO =0
= 2cos20 * 3cos0 - 2= O secO = cosecO
coso:-# 0=45"
* cot45o
tan45o
=1+L=2
coso= -Bt5
4 r77. cosec'18o---]-
cotz 72"
coso= -3+5. -3-5
'4' = cos""'18'--j-
tan2 18o
2-8
4'4 = cosec2l8o - cot218o
=l
c,
178. cr*0=90'
-1 cr = 90'- 0
2'
F = 90o- ct

.o.e:1 (1
- sin2cr) (t - cos2cr) x (t + cot2p) (t + tan2p)
2
= (cos2cr, . sin2a) (cosec2B . sec2B)
sec0 = 2
1
cos'cr sin2 cr
172. tan22)"=v cos'B sin2 B
2

px =
cos'o sin2 o
b1 .oJ(go'-o)' .i,r'(go"-o)
rt =.[ * r' cos'crv-- sin2 cr
. o o
I

sln'ct,^ cos'c[ -r

sin221'= [sinO=cos(90"-0)] 179. tanO * cot0 = 2

2

= cos671o = tanO *-L -=2

tan 0
2
+ tan20*1=2tan0
. =) tan20+1-2tan0=0
cos67!'=
2 -S r' :+ (tanO - 1)2 = 0
'lL+
+ tan0=1
173. sin21o + sin23o + sin2b" + ........... + sin2g7o + sin2ggo 0=45o
sin45o + cos45o
= (sin21o + sin28g") + ............. + (sin243" + sin247.) + sin245o
116
:----T----=\Z
-22-[#)' rl2 ,12
181. coslo cos2o cosSo cos100o

-- 4r
 = coslo cos2o cos3o cosgOo ..........cos100o = cos2e + (cos2o)2
= coslo cos2o cos3" x0x cos 100o = cos2e + sin20
-0
183. 3sin0 * 5cos0 =5 .......(il ,185. tanlOo=1tan15o tan75o tan80"
5sin0-3cos0=x ..... (ii)
= tanlOo tan8Oo tan15o tan75o
(i)2 + (il)2
= tanlOo cot1Oo tanl5o cot15o
(3sine + 5cos0)2 + (ssine - 3cos0)2 - 52 + x2 =1x1
+ 9sin20 + 25cos20 + 30 sinO. cos0 + 25sin20 + 9cos20 -1
30sin0.cos0-25+xz 186. Same as 155
187. sin7x = cosl1x
+ 9sin20 + gcos2o + 2bcos20 + 2bsin20 - 25 + x2 + sin7x = sin (g0" - ttx)
=) 9 (sin20 + cos20) + 25(cos20 + sin20) = 25 | x2 = 7x=90" - l1x
g+25-25=x2
= =) 7x + l7x = 90o
= 9=x2 + 18x = 90o
x=*3 . --tro
L-U
184. sinO + sin20 = 1 ffi,
sin0=1-sin20 tan9x * cot9x
sinO = cos2e tan45'+'cot45o
3f,6[, =1+7=2
cos2o + cosao
 cos5zo + cos68o t cosl72o t5't ITFT Elrlf- I
(a) o (b) 1
(c)
4
(ils 1

1
(c) -1 (d) 5sin0+3cos0 _?
2 11. qR 3coto =4E), d 5sin0-3cos0
tan7Oo = ?
(a) tan5Oo + tan2Oo (b) 2tan50o * tan2Oo
(c) tan5Oo + 2tan20" (d (a) (b) e
2tan50" + Ztan2O" ;
( tan35'
cot 78o sinl60o sec4Oo -)
3' [.",t"* t""tT *.orzff* I
.".14f -l.J fl
-\
rTIr B- (c) 3 (d)
,
(a) (b) 12. qfq 7sin20 *3cos2e =a€flh e Bfi
3 4 dq + iG[ cotze H1
J5 rII;I .FII EFff?
(c)
T (d) Eri t mit =rtt
qfq .r=asece +btang,y=bsece +atane d, d (a) 3 (b) 1
4. 7
*2-y2 =? 3 2
(a) 4ab sec 0 . tan 0 (b) az -b2 (c) (d)
i 1
(c) bz - az (d) a2 + b2 13. qfE (tan 3b' . tan 55") = sin A, d lA=
fr (t"t 0 * seco) =x,E), H) tanQ - ? (a) B5o (b) 550
(c) 9oo (d) 1900
L4. Isin6 *cos0]=?
I
(a) =O (b) <2

6. .ffq (trr,R - tanB) -, eilf (cotB - cotA)= y E), a) (c) >2 (0 <J,
15. qft 4z = E), d AHr rrrr Hr Etrn?
(b) (x- y)
1+ tanl
(d) (x+y) (b)-
l-tan-i
2
7. ( 1+se1e -cosece) ( f+tanQ*sece)6I qFT qtlq{ Etrff-
(a) J, (b) 4 (d) secr+tau
I
(c) (d) 2 11
, 16. l+tan., o+-l;;;E'il qH nrd s,t-
sinA-sinB cosA-cosB
+ I
cosA+cosB rt141sinB= (a) (b) 2
(a) sinA cosB (b) cosA cosB ,
(c) tanA tanB (d) 0
2
qR r=o,E), E) cosx *
L=,
(a) 1.5 (b) L.2 L7.
(c) 1 (d) o
3 . . l-cosO (a) (b)
30
10. sR tan o =; Ei, d ;ffiol qt;T Erd Ht- * 16

160
(a) 4
(b) 5 (c) (d)
s s
* 3
 (c) B.lb (d) 4
4\\
18. qR tanQ = . 6t,dt 27. qR A-B=9ood, d cos(A-B)=?
(a) J, (b) 0
)
(a)
l (b) -tI (c) o (d) 1
sin 70o + cos 40o
28.
(c) I
l (ili J
cos 20o + sin 50o

I
(a) (b) .6
te. s tFI r1-{ {drE ve -j$3 -o= = 3
cot' - cos'0
0
T
(a) go' $) 4b" -_? I
(c); (d)1
(c) 60o (d) 9oo z

20. qR sin (A *D=+ *r sin G- B) = iEi, d 29. fr sinzg -2cose .I =od, d 6 tFI rIFI qdr{,
(cos2B- coszA) = ? qd o'< e <90'
I 'It
(a) (b) 1 (a) (b) +
, , o

(c) o (d) J, TE 3n
(c) (d)
, ,
i, *
2L. (4 cotz lu rTFI IFTI
!* - qinz dtnz 30. fr sine =0.6d, d (tang * cos e ) qq1tp dtn-
"""'
(a) 1.bb (b) o.gs
5 t7 (c) t.7o (a) 0.64
(a) (b)
; 6

(c) t9 31. qR q+ *r *tm- +r tr sNil { Esm'r qrq fsaqr d'nZ

+o (d) ;
o (a) t4bo (b) 7b"
sin0-2cos0
(c) 108" (o Lzo'
22. fr tane=ld, d sin0+3cos0 ='l 32. qR 8 tanx = 15 d, d (.ir* - cos.r) = ?
(a) -i
I
(b) 1
(a)
i
8
(b) t7
7

(c) o (il. + 1 (c)

n
1
(d nl-

23. sin 0 zIT cos g tFT sTfit-frTq rTH Tfrri- I

(a) J, (b) 1 frcote=Ed,d
(c) o (d) -1
,)
24. qR sinzg+ cosec2g =rd, d x =? (a)
(a), <2 1 E
(c) x >1 2 5
(c)
m ,
fr cosO = -n E[, dT tan0-?
sin 30o cos 45o
_r)
34.
tan 60o
m+n
(a) (b)
.

(c)
n2
2

*^2 (d)
(a)
n
J6
(b)
r 2
m (c) 2J, (d
l.\ '/5
26. qR coso = 35. qR
t 6l,dt sing +tang -? x = sinA cosB, y = sinA sinB, z= cosA,
qH HI d.nZ
x2 + y2 + zzE;l
SJ' (a) o (b) 1
(a) (b) 3.5
3

-- U+ --
ED lv
 (c) 3 (d2 (a) 1+ sinA (b) ( r + sinA) (tanA- secA+ 1)
36. qR tan245" - cos260o= x sin45o cos45o tan60o if{ x= ? (c) 1 - sinA (d) g+i't sti Tfr
1 J secx+l-tanx
(a) (b) (>r
, O 't
tanx-secx+l
l+cosx-sinx
(a)2 \4,'
sinx+cosx-l
--
1-sin.r+cos.r 1-sinx+cos.r
(d)
37.
\U,'
cosx-l ' sinx-l
(b) tan2 e 47. qR tanzB = d, d e +"28=?
l+L#
I
(c) I
(il sinzilJ; TI
(a) ; $));
TE

t
z
38. (cos g - sin 0 )2 + (cos g fsinO)2=?
(a) 1 (b) 0 (c) - TE
(d) t;
4
(c) 2 (d) .6
s9. qR A+ B = 18oo d, d sinzA+sinzB=? cosec 0 cosce 0
48' cosec e-1
+
cosec e+1 =
?
(a) 2sin2A (b) 2coszB
(a) (b) 2cos2
(c) o (0 J-z"or'A
2cosec2 6 g
(c) 2cotz g (d) 2sec2 e

40. qs' nd, c =2,b=2,sinB=

f A zA*n- 4e. qR cosQ =
; (".*) d,d 2cosag-1=?
(a) 3oo (b) 45'
(c) 60" (d) 9oo
(a) (b) 2a
41. sin2 78o+ cos2 78o =? "* *
I
(a) (b) .6
, (c)
:(r.;) (d) a2 -+a-
(c) (u)+
1
50.
t*.,] [#-r] - tan2A?Fr rTFT ffi q$R *nu
1- sinA (a) 2 (b) 1
1+sinA (c) B (d)0

(a) -+
COSA
_ tan A (b) cosA- I
tan A
51. fr sin20 - 2cos0 * i =0 d, d 0 $1 rllq HI d{t,
ll
(c) - tr"A (d) cosA
- tanA wt (r<e<;) n
"".A
1.. It
(b) :It
43. HI( tan O= , fi{ tan g
] a, d 6 + e= (a)
2 5

(a) lt
i (b) 0 (c)
TC

4
(o=fio
(c) TC

i (d rE 52. qR sino - * n, n
cos2Oo . cos40o. cos60o.cos3g" ffi q{rfl +? (a) 0 (b) 1

22
(c) cosu (d) srnu
(a) (b) 16 *
*
53. qR 2sin2 x+ cos24bo = tan4bo ertt ( x < g0" d,
(c) 8 (o: d tan-r = ?
O"

45. (tanA + secA - 1) cos A = (a) 1 (b) Ji

ii:l
:i::l:

!:ii!
 II
(c) 3 (a)4
(c) --E (d) 3 n3n5n7n 9n
VJ 63. cot cot cot cot cot =?

b4. qR cos4A = -;d, d cosA EFI tfidil qH drn? ^ ^ ^ ^ ^

I
(a) -1 (b)
(a) 8 (b) 6 ,
(c) 4 (a)2 (c) o (a) 1

64. qR tanQ .tan20 = ld, d sin22 g +tan220 = ?

tan2 60o -2tan2 45" *.""2 3oo
55t l=, 3 10
3 sir2 45o.sin9oo+co"2 60o..o.3 oo l (a) (b)
i T
49
(a)
n (b) 1
3 (c) 3i (a)8
t4 4 65. 2sin2 6 + 3cos2 e tFI qTirrt qrq iliT fifqq-
(c)
s (d)
t (a) o (b) 3
(c) 2 (a) 1
8
fr sine =; ,qd o"< e < 90"d, 66. tan 4o . tan 4So..tan
(a) 2
4'7" . tan 86o
(u)g
= ?

d t"., e + sec g tFT HFT +Tf Etrff ? (c) 1 (d) 4

I 2 tan0+cot0
(a) (b) 67. qR -2(o < e < 9096I, rll sin6=7
t l ffi
4 2
(c) (d)t 5
(a) (b)
l E +
ll I
57. qR r+:=2cose EI, dl ,3+3 -? (c) (a)
xr , 1

(a)
].o,,
(b) cos e Gs. qR o< e <eoo.ft =2d, d o =?
#.#
(c) 2cos3 g (d) 3cos3 e (a) 3oo G) 4b"
(c) 60o (d) 7b"
58. fr co.1a-B)= j.+, sin(A*e)= jd, 69. qR 3sin2 0 =2sin30 +{ 0o< e . r, El, d .o. 0 =?
d z A4it {i-dq EIrcq-.F' qH HI dtnZ (a) o
(a) lgbo (b) 60'
(c) Boo (d) 1ob' (c) -Ll4
70. qR c, =cos46 -cosze d, d t{q { t +h-qr sg +?
I sin300otan240'sec(-390") I
59.
L l=
(a)o<a<1 (b)-1 <a<1
(a) .[ (b) J, (c) -;= a< o (d)0<a<1
2

(c)
2.6 (a); l.\
E qR .,
21. +g=
] .*r sincr =
t 6[, ill sinp =?
60. sing+sin( 0 +120')+sin(e +240")tFI rIFT Hr drn? t;
(a) o (b) 1 (a) \z /L\ 2J'
\L"
3
(c) 6 (d)2 3

61. qfq 2sin2 0 + cos245o = tan4b' 3t{ 0o < e (c) 2

(u)i
a
sin6=r
(a) 1 (b) .6 72.qRogO< n etf tanQ =cote =-1 d, d o =?
2n
(a) (b)
(c) I
(o: T +
JE
5n
62. coszg ( 1 + tan2g) + sin20 (r + cot2e) - ? (c)
o
- (u)+
(a) 1 (b) 2
 2-2tan2 A \ \
73. zTR =lEI, c[T 7 A=?
;;A
(a) 45" (b) goo
(c) 15o (0 60o

24. qR tanQ = | .tr oo< e < goo d,d 83. qR sin6 + sin2g= 1d, d co.20 * cos4gffi q{1-q{ +?
(a) 1 (b) 2
sin0+cos0-cot0 _? (c) 3 (d) 0
cos ecO - sec 0 + tan 0 84. fr .o.g + sin O = J, cose, cosg-sing4'T ril;T dtff-
(a) 231 (b) - 2t3 (a) 2sin Q (b) f sin s
377 3n (c) sinz g (d) sing
231
(c) (d - 827
w ,,s2
75. qR tanQ -cotg =cr*{.o.g +sine =bd, (b) sinz e
d (u' + 4) (bz
- 112 ffi q{rqr +-
(d) tanz e
(a) 1 (b) 2
(c) 3 (a)4 8G. qR sec 5A= cosec (4+ 86'), qd'seg+'yI*. qiur t' aqr qn
76. qR x2 +yz = 1d, d .,rytfi1 qFT Hr drn? {rd 6,t-
(a) 18o (b) 27o
(a) ll (b) 2 (c) BG" (d) 9o
,
(c) 2 (a) -2 87. qfR tanO + cot0= 2, if6[ tanraO * cotraO tFT q-{ Hf + ?
77. qR sinQ +cose =,[ sin(90"- 0),cotet5l ril;T Ff6,1d-
(a) 4 (b) 2-n
(a) Ji *t (b) Ji (c) (0, 1
-r 2
(c) z (d J,
(1+ sinx)
7S. qR tanQ * cot O =2,tan26 +cot2g 4'T rIFI nfd 6t- 88. qR o'=
(a) 1 (b) 2 ffi, Er ai Aqrr+{ +-

(c) o (d J, (a) secx - tanx

2,"n2++mt,
79. fr cosecA - Jr, d,' d ! $r rrFT nrd m,t-
4tan" A- cos'A (c) 1+sinx
.

a
4
(a)
=)
(b) -
J

4
8e. qR , #= q, rrer x(.r qr+iur t, rq l? -Ptan xtrr HFT

Hr +?
(c) 8
(d)=I (a) p (b) q
7 I

15 \ \ (z
+ zsin o)(t - sin o)
(c) tD
80. qfq cotQ =; EI,ol (1+cos oi ao dfqq- pq (d)
q
ilQ-2coso)
90. 4 sin2x + 4 coszx z5I rIFI dtn?
(a) 225 (b) 39
(a) + * qnqr qr 6q *
64 64 (b) [2, +]qr t
(c) z * qtrEt qr sq + (d) 4
(c) t25
64
(d) 321
64
el. qR ("*F)=f, d, d (r+tancr).(l+tanp)tFI rIFT q{rq{ +?

d
81. qR x=asin0 efu v=btano d,'xoyo +-+ffi rII;I {kt
(a)
1
(b) .6
6t'-
15.
' (a)1 (b) J'
(c)1 (a)2
(c) o (dB 92. lsin x + cos xl -?

_-47 _
 :i:::,:::r:::i:::1i::::::::r,r,]t,ti:::,:ii:::tl:rii it:

1
(a) r d qnqt (b)lt sTftr-fr
(a) =O (b) <2 (c) rtqq (d)2dqat
(c) >2 (o .Ji 101. slnz 0+ cos' 0 + sec2 0+ cosec'e+ tanz 0+ cotz g fiI g{i[IT TTFI

t+r+r tz
gg. qR A+B=90",(lo[ tanAtanB+tanAcotB sin2 B (a) 1 (b) 3
SinAsecB cos'A (c) 5 (d7
_?
I
(a) tan A (b) sin A 102. qR cota rtana =-H -cosa =ndEl
cosa
(c) cot A (d) cosec A
(a) * (*r')+ - ., (,r-z;a1 -1
fr A, B w c fiTe ABC d era'stor d, d tu., T =,
11
(b)
'"(-2")s -"('""2)B -1
A
(a) cotA (b) cot
7
11
A
(c) tan A (d) tan (c) *
" (-"2 )s - (.'*2 )' =
t
7
9b. (tan35"tan45otan55o)Uf qn Effi{r +2
(d) fli't st$ rd
2sina . 'l-cosa+sina
----- --:-
(a) 1
(b) 2
-ruo. ffi l+cosa+slnd =v
- rt-T
l+slnd =i
,
(c) o (d1 (a) ;
1
(b) v
96. fr sec (ze + 28') = cosec (80" - 3e) d, d e qr qH fsr{r -v
dmz (c) 1-y (d l+y
(u) 8o (b) 5" sin2y 1+ .cos y sin y
104. 1-, +
(c) 60o (d) 9o 1+cosy slnY -. r-cosY =-!
97. qR e gfi tr{rffifi qr"tur d, e+{ :*rn2o tan B0 = t d, d (a) o (b) 1
(c) sin y (d) cos y
(2cos2 rTFT frnZ 105. qR sinp Hcosp IItrS-{ut a)cz -bx*c=0* W *, fq
T-rrr51
(a) az +bz +2ac= O (b) a' - b2 + 2ac = o
(a) -,1 6) 1 (c) a2+c2+2ab=o (d) a'-b2-2ac=O
,er'Oo cot30_ _ 7-2sin^20.co^s20 _,
,.0e.
rvv' *
(d), 1 - sino.coso -
I
(c) o 1+tan20 1+cot20
(a) 1 (b) 0
(c) 3 (d) 2
9s. qR sin 17'= { d, d (sec17"-sin73")qT qFT t+,ilfl dtnZ
v qR T, =
107. + cosn e il-{ * T, =-:,
x'
"in'0
(b) ----
y{y (a) sin B . cos B (b) sin2 g . cosz e
-r (c) sinz p . cos g (0 sin Q . cosz g
(il;wv' 108. qR cote (L+ sin 0)= 4mH cot e (1-sinA)= 4n,Gl
(m2 _n2)2 - ?
(a) mznz (b) msna
99. qm'qr*tur Egu xYZt, d vqr qr*tur +, qR xY= zJ6 (c) mn (d) m'n
iTeTI XZ-YZ=2d, d secx r tanxtf,eHf *nZ (t1\
loe' sin' s ' cosz o =?
(a)
1
(b)
[;Po -;;%*;;4;;;5]
-vb G
1+ sin2g.cos2CI
(a)
J6 2+sin20."or2e
(c) zJG (d)
2
L- srn2e.cos2e
100. fr oo<0<90'd, d sin0+cos0 6T TTFI flr drn? (c)
2+sin20."o"2e

-- 48 --
S.S: C. ADVANCED MATHEMATICS
110. qR secp + tanQ =2d, d .e.A c[1 rTl-;T iFII trfrn) 2\A
L
(c) 32
7 {ffi.
(d 82+ffi
(a) ; (b) 5
b 120. gm, 100trfr dn Htcr (qrwq) Arq q+i rrE 6i"r E1 ffi
d qls HI +, qR {ss'r qll $Trrr +m{ $kTr Ee{T 1sffi ql
(c) ;5 (d) J, qN ffirdr *?
4

111. ?TR cosec0 -sin 0 =ldl4l secg -cosB =md, d F*' (a) bo43'38" (b) zo4g'gg"
(p + m, + B )th.t q-{ HI drnZ (c) 5o34'18" (d zo34'18"
(a) -1 (b) 0
121. zTR gq. trgfrq Hl sirr qrrr +ffi 61rdr EsrT b0 rtfr q,r qm
rFtIfiT t nqr f{e11
(c) 1 (d) 2 ft w 60"8r etur rrrar t, d trgf,q o,i
7\\ r e\ ( zz \
112. qR cosec 0 -cotl =, d, d .or". e 4I q;I IFTI drn? ilqri t ["
=;uJ
47 51 (a) 48.72 +fr (b) 45.72 tfi
(a) (b)
n 28 (c) 42.72 tfr (a) 4b.22 tfi
53 49 Lzz. L*rfr{ Bofr{e q{ € + fr qidi * fiq ?fiT rrrq ifgq{
(c) (d)
qEt
28
o
^
113. *r qH (a) 3r rad (b) 2tt
rad
;radtfi'l 7 5
(a) 84"28'tg" (b) 36o28'19"
(c) B6o2L'49" (d) 84"21'49" (c) 5n
.,rad (d) 5r rad
7
114. qR lrad= 57o76'2L" d tOradtF'T rTl-;T Hf +? 123. q5 qfdqr 101Hrs it z+ER TUh +tar tr qf6qr Etrr +
(a) bzoo16'21" (b) b7Bo4B'10"
110 tBqT {ti{ *.d d errtrtr tfif,{r sqq trn?
(c) 57Lo4z'4o" (d) s72"48,8o"
(a) b +irs (b) 7.b Q$rs
11b. ?TR ffi Eil"r s1 !m.r$ 2s"46,ss" d d Ts stq *1
q+. s
(c) 10 t+rs (d) 81ffis
q+r$ m qH drnr
(a) 148ob4'Bb" (b) 146ob4'Bb" rz4. 6otfr @ sl g.F"ettrfi, Btfrilr EoTT 18tfr qr sFr rFnfrr
(c) r49"s4'25" (d !47"44'Bb' *t ileqn, E{r FFr-{ d Etur q+rn stffi1 qrq drTqrr +t
116. qR ffi q+rr 61 go {f,r$ 1bo49'b0"d d yq qt"r +1 roo
(a) 1bo (ul nf,
gori m qn drnr
(a) lbSooBo'20" (b) Lb8z"B'20"
(c)1b8B B'20" (d) 1b81 go'20" (c) 20" (a) ,r;
112. ?TR q+^ qF6qr oTrri qt { goqffi{ trrnf,T * d qo qfi Ffiz Lzl. b4tfi trqr erd gf, $T qrq qfq +q qr 2o"rfil etur qrmr
d f*,r* fufr or et"r rq q,t'nz
(a) tzo' (b) 7200 tdqrqqlffiHr+? (, =?u\
(
7)
(c) 1080" 540'(0
t
11s. ?TR qfi qF6qr qfi q) zzo+rfi{ eqrm t il qa q+' ffirs (a) 1,^
(b) 1774 rl{t
\A
{
t*m, tGr{ etur rq q.t'n?
1e
7 {t{I

(a) 7Trad
U
(b) 2r
U
rad (c) ,tf m (al 1e; ffi
qtr Tf,rfiTr r.sfi qr furc t
126. vtirTqfr (s) 3Tt{ 3{r{q (A)tE,,tqt
(d) 4r rad
3tt
(c)
; rad
u d qfi fr{rd f{S eo'q ffi TrTt *t qf( fr * qr t
q{
g*. qru 3sffi. tr qr d sotfr. frqr qq sui 6nr f€R fq5 +t {fr 100frar + d +d + qt + frq *1 ss+'
11e. Tfl 6r
trq Efi qri qrd {ft (errwr) +
q{rq TE q-qrq E1 @ + (a) 628 frat (b) sr+ fra
(c) 1b7 fiar (a) 2gb.b frat
(a) ,r
f m. (b) 84 tfr 127. gq6 6q.FR 3sfr{e t 4q5. s{q ilfr'q+ efr.ffi q{ Enn
ilq E1 G qiitr +1 =+rq t

-_49_
S.S.C. ADVANCED MATHEMATICS
 (a) 27oo (b) z+s" rgs. .lR o < e < q <90",d firqfdfuf, { t +{-rn q.5' sfr +?
(c) 255" (d 26b" (a) (sin g * cos 0)'> 2 (b) (sinz B * cosz q) < z
(c) (sinz B t cos2 q) .z (d) (sinz B + cos2 q) rz
r28. rTr;TT t+ * OT qrq \ri' iern * do o. (i)'* *
rso. qR o < e. ; 3+{p=secza t, d trqfrfiq-a t t eh-qr
q"Trdr tr iro t
61 *q tor+ qfr w r rtfr EIRI qrelT

f{rffir rsr qri ilt+. fuffir =fq Ei qfr cca t*, dr qo w1 tz

(a) p<1 (b) p=1
(c) p>1 (d) p>1
(':Tu)
rsz. fr o< 0. ; fr cosp + Ssin 0 =2d BqT q-{ F1t
(a) rrfrm (b) rofr .+*
tz
5 \A
(al ,rfr.+* lf
(c)
,, ${l
16 (a)
E
(b)
i
12e. Hq ffi 6qI + qrt'*t {Ti i fd d z+q"} errfr tr If lt
(d)
+ q") 12 fr{e t qa f6q E}"r Q ffi'r
(c) ;
b ,
(a) 63" (b) 64o 188. qr-{dfqq, yd Rtf,qI el ,iler *t +o"s ertwr d' gfl E1
(c) 6b" (d) 7oo r\^
LIEII FIT 6i
13s. FrqldEd {t +{-qT gm'wfr tz (a) R cos 40o (b) R sin 80o
(c) R sin 4o' (d) R tan 4oo
(a) tanx> L;45o(x<90o (b) sinx)|,0".x<30o
rgg. FrrTldlqf, qffi Q qqf*6 mtrqi qr frqn +1fqq:
1 sinsg +2sin2g +3sinB
(c) > 60'< x > 90o
1. ffi 0 e R* fflq {q dqfi q,r erftr+-oq qn o*r
cosx
,;
(d) aoo< x< b"d r* ffi qn * fr( sinx=cos.tr 2. ffi g eR+ldq{qfd*..FIqi orddv+mr
rgr. fr sinx=.o. y $1 qtq ,$ivr5"lur t, d , utt vd wt t
w{ffi *.eni d +{-qr / vfr trtz t
+nq tu (a) *+q r 6) *+e z
(c) tir*tz (d)rdr*tqfrz
x-y=ilt
3tt
(b) x+y= Z
rao. qR, - * ].o.'e,d
lt f,sin2e
(c) ,+y= i (a) , *r= I (a) 11 (b)P>;1
E<P<,
182. ?TR 0 < r
= ; +, t trqFeFqo { t +{-' T qm' qtq sfr
tz (c) 2< P< B (al -# =i. #
qR seca de[I coseca qfrq-{q *' -p** q =O* 1[({ t d
(a) sin', <
t etR
"o.',
,l (b) sinzx,
! .+. .o"'r.
I
141.
(a) p2-p+2q (b) ,z=p*2e
(c) p2 = q (q+ 2) (4) qr=p(p+2)
142. qR seca dstrT tanQ {fi6E{uf ax}+bx*c=0d-aetd (r,
(c) sin2x. .+.
| "or'r.f, b +0) seco - tanQ ST qH *
(d) t oq-t-eq ger', 1t da tr
sin2x, cos2xd
a
rsg. ffi p = tanzx + cot2x*, d f+Tfmfuf, { t +{-qr qfi sfr (a) -r
D

tz
(a) p<2 (b)p>2 (c) 1-;a
(c) p<2 (d) p>z D

184. qR 0 <x< 4b' o+t 4b" <y ( 90o,6 StqFobc d t +Yq-qr 143. qR x=h*asece He]I y=k+bcosecg d
qo vfr *z
b2 bz
^2
(r..h)z - O*qz- =L
(a) sinx=siny (b) srnx(srny (a) ^2 (b) +
(c) sinx>siny (d) sinx < siny 6 1r-oj =L

NCED MA
 (c)6 (d7
(c)
+*u4=,
.fR sinA-
(d
ry-Q$=, 148. qR
d'nz
cosz a rcosz 0 =2d, d tan3a *sin5 B qI qn flq'mf

f aa. .fi cosA = fi cosA,d cosA+ ,16 sinAtF't

rTH +
(a) -1 (b) o
.f A
(a) sin (b) - f sinA
(c) J6 cosA (0 f cosA (c) 1 (d) +
V,J

145. fr sinB defi coso {IfrE51TI +bxr c = 0* y62

ae t d 149. (sec x. sec y * tan x. tan y)' - (sec x. tan y t tan x. sec y)'SI
(a) (a - c)' - b2 - cz (b) (a -c)' = b2 + cz v{mtr qH +-
(c) (a+c)2-b2-cz (d) (a+s;z=b2+cz (a) -1 (b) 0
146. sin (cosx) 61 3fq-5-aa qrq + (c) sec2x (a) 1
(a) sinl (b) 1
^
150.qfE o"o"20 o =SilslT oo< o <good, d qFT
(r) r J5) cot'9 - coso9
et5T
(c) .t"[16.,J (d.t"[+.J t*,dqt d'nZ
(a) 3o' (b) as"
L47. sinze + cos'e + tarrz7 + cotze + sec'7 t cosec2 e 6l
qrdq qH Hr Etr[, qR o <a <go" *t
(c) 60o (a) Eri t Eti Tfr'
(a) 1o (b) 5

1. Cos 52'+ cos68o * cosl72"

tan 20u + tan 50o
) . cos(tz2'-52')
(t72o +52"
- tan 50". tan 20o
= cos68" + 2cos
1

2 2
= tan7Oo - tan7Oo. tan50'. tan2}o = tan2Oo + tan50'
=cos68o * 2 cos60o cos112"
= tan70" = tan2Oo * tan5Oo I tan7Oo . tan2Oo . tan5Oo
tan7Oo = tan2Oo * tan5Oo * cot20' . tan2Oo . tan5Oo
x;zI =
= cos68o + 2 cos112o tan70" = tan2Oo + tan50" t tan5Oo
=
* cos 112" .'. tan '10" = 2tan50" + tan 20o
= cos68o
( tan35" cot 78o sin 160' sec 40o - )
(68'+l 12') (ttz' -68" ) .{l-T-T-T-_Il
=2cos Z .cos \ cot55' tanl2o cos20o cosl40o )
2
=2cos9Ooxcos22o ( tan35" cot 78o sin 20o sec 40o \
_ t_f _a_r__I I

=2x0xcos22o=0 (tan 35' cot 78o cos 20o -cos40o )

2. tan 70o
= tan (so" + zo9 = ( 1 + 1 + tan2oo- sec24oo-1)
=1*tan20"-sec240o

*..**-*""........"..-."*.......-.:.'I'1..::....
S.S.C. ADVANCED MATHEMATTCS
...... ".............*- , *
 * **I!*{t w*}#w **$ffi
4. ,=aseqo +btano /sin0+coe0-l\ l,cose+sino+l\
= l=(aseco +btano)?...........( = l. "r"e I t """e .l
y=bsece +atane
sin20 + cos20 + 2sin0.cos0 - 1
+ y2= (b sec 0 + a tan6)2..............(ii)
sinO.cos0
(il - (ii) -o
*2-y2 =(asec 0 +btanQ)2-(Usece +atane)2
sinA-sinB cosA-cosB
= a2 secz g + bztan'e + 2ab sece. tane- bzsecze 8. - slnA+slnB
- a'tan? g - 2ab sec e . tan g "o.a*aotg
(sinA - sinB)(sinA + sinB) + (cosA cosB)(cosA + cosB)
= a2 (r) + U2 (-r) -
(cosA + cosB)(sinA + sinB)
= a2 -b2
D. 6q drr qHfr t f*, sin2A + cos2A - sin2B - cos2B
1*tan20 =sec2e (cosA + cosB)(sinA + sinB)

-tan2o = 1
.'.sec2g -0
+ (secQ - tan0) (sece +tan0)=1.............(i) ,2
cosx*
t<qr gen t -
sec e + tan 6 =x .................(iil x tfi-t rli rqi qt
s+fiq (ilt 0
cosO +
(sec 6 -tan0) xx= 1 7
I =1*0
(secg -tang)=; (iii) -1
- (iiil 10. tan0 _p
(iil
-b
I x2-l 3
2tan0 =x- -xX
4

*2 -l [=
...
"2x .h-
(tanA-tanB)=x
(cotB-cotA)=y /
1-=
11 1-cos0 5 1
< tanB tanA --Iv r ................-=lt=-
1*cosO 1+l 9
5
tanA-tanB
= t"rrAjrrrB =Y 11. 3cotQ =4
4
cot0 = 5
= ; = ##i-j# x (tanA' tanB)
5sinO+3cos0
x
. -= tanA. tanB 5sinO-Bcos0
v
5sin0 + 3cos0
l+tanA.tanB
cot(A-B)= tanA tan B sinO
- 5sin0-3cos0
sinO
x+y
xy tr*gr 1
5+3cot0 3
7. ( 1 +cotS-cosecO) (t ttang +secg) 5-3cot0 = 5-3x 4 -g
-3
( cosO 1\( sinO 1\
= ' ' ' L2.7sinze +3coszg -4
[^ sirrO sin0J [^ cos0 cosO/
= 7sin2e +3(1 -sin20)=4
= 7sin20+3-Bsin20 =4
 * * ffi wg* @w*
+ 4 sin"g =
_o
1 sinO - cotO
I
' 2tan0
.'. sin Q - , 43 16-15
p_1 54 20 13
h- 2 =zx4=q 20'- 8
33
=!6 3
160

cot2s = (:)'= [f)' =3

18. tang =
4
-
t1
13. tan35o. tan55o = sinA
sinA=tanS5o.cot35o p4
sinA = 1 "b 3
sinA = sin9Oo
2 A= 9Oo
h- Ji\82 -5
L4. '.' asine +b cosS ST 3ftltF-6q qPa

asinQ +bcose til q]-dq rIH 1

= -Jw E
=
b 1

3
sin Q + cos Q z5[ IFI = +r[az ag =
b

= +,ffip = +J, ' .o.2e

It. - o =,J
= .Efrnr cot"O
o---------
- cos'0

= cos20 = 3cot26 - 3cos2g

15. A2 = (r
- sirx) =3cot20 =4cos2g
3cos20
(r+sirx)(r+sin-r)
A2= (t
- sln-r) (r + sin-r)
= ffi =4cos20

q
+ .i.o;2 =) sinz e - I
.+
1r
A2=
(t - sin2x)
+ sing - !+
t- E6
- -
_

+ si.o)2 2
.A2 1r
=
aoa2, 0 =60o
1* sin-r 1 A
[= cos.r
= cosx
sin-x
cosx =SeC;+talLr
20. '.' sin(A + e) = -J'
- + sin 14 + B) = sin45"
1 1
---- + B = 4bo .................(i)
^v' 1+tanzo
r(,. a T-
1+cot2o
... A
1
... sin G_ B) =
sec"
oT6
0 cosec" 0
['.'r+tan2g =sec2e] O
.'. ({-8) = 45".............. (ii)
h+cot2e=cosec2e] Hfrfi{q (il aqt (iil t, rrTdrr qr-fr t
=cos2g +sin2o =1 2A = 90o
3 .'. A = 45o
t7. cos= ilsll B = 0"
5
Sltt Cos2B - coszA = cos2Oo - cos24bo
b3
h5
.'. P= ,f5z-E =a
(r)2
-
 ii,i.t: itii,l,ilil iiiiiiiiiiii :ii:iiiii:ii:tiii:iii:i:iiiiiiiiii::iiri:ili
11 1
_
-'1-22 cosO =;J
b1
2L. 4 cotz
f .""'
+
i- "r"' i .h3
. 18oo 18oo
=4coto B *sec'" U -srn'"
18oo ... p= tr=t = J3z _L2
n
=Jt=zJT
= 4 cot26oo + sec230o - sin245"
.'. sin 6 + tan S
(r)2 (z\2 (r)2
=4x trj .[rJ -tr) bp
-h'b
t4 1
2J, 2J,
4" 5 * T-
5 2 31
8+8-3 13
? _ 8J2
3

sin0 - 2cos0 27. cos (A - B) = cos90" = 0

sin0 - 2cos0 cos 0 sin70" tcos40o
sinO + 3cos0 28.
sinO + 3cos0 cos20o +sin50o
cos 0
sin70o *sin50o
sinO 2cos0 sin70o +sin50o
-1
cos0 cosO tanO -2 ['.' sin (90'- g) =coso & cos (90'-0) = sino]
= sinO * 3cos0 = tanO + 3
cosO cosO sinzg -2cose * i =o
_1.-2 _ -1
1+3 4 =1-cos2g-2cose*i=O
23. sin S 4T cos Q Eh-l 3Tfiffiilq rTr;T 1 *tf tf +4-4cos20-8cosS+1=0
24. sin26 +cosec20 =x =4cos2g+8cosQ-5=0
4 cos26 + 10cos$ - 2cosg - 5= 0
., f . 1 o^ =
' sln'tl
- srn'U =x (2 cosg + 5) (2 cos0 - 1)=0
=
'.'2cos6+5 7 0
. [.i.,e--L)'+2=x .'.2cosg=1
I sinO/
1
.'. cos e =,
qrq (.,"r-*)' =od, d x=2 + cosg = cos60o

qtr qq [rir,e-- ] \2 '.e -60"=;

\ sine),.omlx>2
x> 2
30. sinO -0.6=
6
fO
m
cos Q
n p_6
bm h-10
hn u= n[\2
= ,fi1u2 = 3
.'. tan g + cos S
p Vn- - m- _!, !=9.,_ 8 _30+32
. tan= b = bh81040
 *=26
62
= 4U
- =1.55
81. ntk{;T= 18oo
IsinO
t-+-
3n 37. "o"012 |

o
tkqq =180x; Icos0 sin0]
32.8tarx=15 [sin2e*"o"2e.l2
t_l
15-! I sin0.cos0 _]
.'. tan x =
8 -b
l-rl2 I
- L"i"e"".eJ - sin2e..o.2e

'
p q-15 8 8 (cos g
- sin 0 )2 + (cos g + sin 0 )2
sln.r - cos _r = h- h - L7 t7 -7
- 17 = (cos2g +sinz0-2cos0 . sin0)+ (cos2g + sin2g + 2cosQ.

1b sine)
33. cot0 - ----
./g- -p =1*L =2
39. '.'A+B=180'
A=180"-B
.h- sinA=SinB
sin2 A + sinzB
= J3+1 =2
= sin2A + sinz A
sin2 e = 2sin2 A
1- cos2o r E\2 1
= -E
2-JE = 2-t
;l 40. sinB

2B
!Z =
- 45"
sin45o

3
AB=AC=2
L3 .'. ZB- ZC=45o
=^ B -R
-.J ... Z A= 180o - ZB- Z C
2-"4 4
b
= 180o - 45" - 45" = 90o
4L. sin278o+cos278o=1
sin30'.cos45o ['.' sin26 + cos20 = 1]
34.
tan60o
1- sinA
1+sinA
(r - sinA)(l- sinA)
(r + sin0(1- sinA)
1GJ6
2'16 J0 L2
1- sinA 1- sinA
==./cos'A
35, xz + yz + z2 = (sinA cosB)2 + (sinA sinB)2 + (cosA)z
= sin2Acos2B+ sinzA sinzB + cos2A tan0 * tand
48. ... tan (e + O) = r_t_r.rerr.4
= sin2A (cos2B + sin2B) + coszA
=sin2Ax1+cos2A=1
36. 1,1 5
tan245" - cos260o = x sin45o. cos45o tan60o
2 3 6
-T-

=
l-ltl
1
- o-r1
=
P
-l

2 3 6
tan( e + O)=tan4bo
1J'
= 1-Z=r, Z
e +0-45"=
TC

4
3J5 44. cos2Oo. cos4Oo . cos60o . cos8Oo
-4= 2' = (cos20o. cos4Oo.cos80o).cos60'
 1
- 7 cos60o. cos60o [cosg . cos20 . cos30 = | cos3el
. 2cos'e - 1 = rl:?. *)]' -'
1111
x
42216 =,li(r *#*,)]-'
= - -x - =
-
45. (tanA + secA - 1) cosA
I .l
= lsinA
Lt** ^-'-.1 cosA =,1+.#.;]-,
= sinA+ 1 -cosA
(t + sinA)(sinA + I - cosA) = ^' *4+1-1
2 2ao
(r + sinA)

= (1 + sinA)
I sinA +
Lr..r* r*.i*1- 1**A]
]. cosA I =+.# =;l*.#]
sinA) t--f +1+ ---L- - + tanAJl
\r +r u'4^/
= (1
[cosecA 1+ sinA secA^+
- - [#.,] i*-']-tanzA
= (1 + sinA) (tanA - secA + 1)
[1 + cosAl [t - cosAl
46.
secx*1-tanx
tantx-secr*1
=
L ."* lL ."* .]-tanzA
1-cos2A o.
1 -,_, sin x l+cosx-sinx = -----------6--t-
cos"A - f,antA
cos.r cos.r cosx
- sin-r_ 1 - sinx-l+cosx sin2A
+1 _ tanrA
cosr cosx cosr = ---
cos"A
1*cosx-sirrr = tan2A - tanzA = o
sin-r+cosx-1
51. sin2g -2cose *l =o
47. '.' tan2B= 1+sinA
1 - sinA (l - sin e)(t - sin A)
(l+sinA)(t-sinA) f ;41
+ 1-cos2e -2cos0 =0

+ | -.o.'o -2coso =o
1- sinA + 5-4cos20-8cose=0
... mnB =
---COSA + 4cos26 +8cose -5=0
sinB . cosA = cosB
- sinA. cosB 15
=
= sinA. cosB + cosA. sinB = cosB .'. cos e =rn - ,
=+ sin(A+B) = sin(90'- B)
A+B=90o-B fi
.'. cos Q = cos,
= A+28=90"= *2
cosecO
* cosecO
48.
cosece-1 .o.".e*1
11 *
= l-sine l+rirre a-b+a+b
_ (t * sino)+(t - sine) =J(^+b)(a-b) =
I + sin20
... eTTrrR = J;2 42
)
=ffi = 2sec20 2a2
=2sece =
I
W "o"e
49 coso =, (.. i) 53. 2sinzx + cos245o = tan45"
 .'. tan g + sec
/r)2 Q

=2sinzx+l--rl =1 Dh8L7255
a4 +--
\./Z 7 =
bb 1515 -=-
15 3
^1
'z
+2sinzx*; =1
57. x+ rx =2cose """---(i)
1

+ zsin?x =
1
1r, ti em qr qd q{
A

1
/ r\3 (2 cos
= sin2x =
Z [,.;J = e )B

sln-r = +
1
,3+{+s, osse
,
0o<x 90"
1
,3 +{ +s
.'.sin-r=;
, =sin3Oo

.'. x = 30o ,3+{ +s e t{fr+,-{ur (il tl

{g fFR, tan x = tan3Oo
e1
1 x"+ g-8c
. t=G ,^
vo

54. cos4A
1 + ,' *4-=
xo
2 [4coss o - Bcos o ]

+2cos22A-r=-l2 ["*+)=2cos,o
58. fflr $n *-
= z"or'zA= I cos (A-B) =
t
=cos22A= -
cos(A-B)=.o, f
t
+ cos2A= , .. G- B) = 60'.................. (il

2A= 30o,330o qtt sintA+g)=

.'. [= 15o, 115o 1
'a

tan 2
+r".2 goo 5n
t""
Goo - 2tan2 Abo
sin(A+B)=sin 7
gsin2 4bo . sin9Oo +cos2 Ggo .aor8 0o
5n
(JB)' -z$)2 +(2tJ3\2 A+B = jb = 150o............... (ii)
= stcrrJr)z .(r) + (vz)2.(r)3
{++,{q (il <qr (iil t
A=105'eftfg=45"
3-z+!
33
7
4 I sin300o tan240o sec(-390') I
=31 7 59.
i 3 .cot(-l 3 5' ).cosec(-3 1 5" )
f
I cos2 10"
24 -.1

8 p
56. sin 0 - t7 h

=15

-- >/ --
{CED IV
 sin22e +tanz2e
= sin26oo + tan26oo
t+*'#)
t+""') =[*)'*(r)'
33
Js"z t^ =i*3=3,
- l,6 x J2- n'
65. 2sin20 + 3cos2e
60. sinQ + sin (0 + 120') + sin (O + 2+0')
=2sin2g +2cos2g +cos2e
=2*coszg
= sin r. [.-(,.?)*.,,[r.+)] occos20<1
erfie {ffq qH - 2+o=z
=sing +2sin0 +n ..o.f tan|". tan43o. tan47". tan86o
= tan4". tan43o. cot43". cot4o
C-D [tan(90'-0 ) = cot0 ]
[...sinc+sinD=2sin+ .cos 2 ,
I
= tan4o. cot4o. tan43o . cot43o
=sing +sin( zr +0 ) -1x1-1
=sinQ -sin0 =0 tanO+cot0 2
67.
61. 2sinze + cos245o = tan45o - cotO
tan0 1

dnr{qrf, a er<trgvre t,
+ 2sin2 g + 2tan9 3
1
1

=2sin20 =1- , sin0 sinO

= .o.o ' .o.e = 3
+2sin2, =l + e =3cos2 e
sin2

=sin20 =3(t-sin2e)
.o 1
+sin20 =3-3sin20
=sln"e = 4

1
.'. sine=,
62. cosze (1 +tan26)+sin20 (1 +cotze)
= cos2g . seczg + sin2o . cosecze
=l*I=2 sinO cosO
68. cosu
^+ slnu
rE 3n 5n 7n 9n -=2
cot cot
a cot cot , cot a sin2e*"o.2e
^ ^ cot63o cot81'
= cot 9o cot 27" cot45" - srn 0.
_ -z
cos 0
-6

= cotgo cot27" . cot45". tan27o tan9o

1=2sin0.cosO
['.' cot(90'- 0 )= tan 0 ] =
= 1=sin20
= cotgo . tango . cot27" . tan2'1" . cot45o
=1x1x1=1 sin 90o
- = sin2 0
64. tan0 .tan20 =1 0 =45o
l 69. 3 sin2 0 = 2sin3 0
tan0= t^n2g=cot20
6sinQ.cosO =2(3sine -4sinze)
=
= tanQ =tan(gO"-Ze) 6sin0.cos0 Z(asinO - +sin30)
g = 9oo -2e 2sinO
+ 3e =90'
2sin0(3 - asin20)
0 =30o = 3cos6 =
2sinO
 +3cos0 =3-4sin2e BC
+ 4 (t- cos20 ) + a cosQ
_2J'
- AC3
- 3=0
+4cos20 -3cosS -1 =0 cotQ--1
(n\n
=cose =
2x4 ITE-- I -- tan- =-1
\ 4) 4

a+.,,FG (n\Tc
= cose = ln-- I =- cot---1
\4/4
3+5 3n
8 4

3+5 3-5 -1
g'g-r:4 73.

'.'0< 0 < n

.'. cose = -14

70. cr=cos4e-coszo
=cos20 (cosze -1) I
+ cos2A=;z
=-cos2e . sin2o
cos2A = cos60o
I =
= -1(2sine .coso)2 ...2,A, = 60"
.'. A =30o
I s
4
(sin2 e )2 74. tanQ -- t2-
-!b
I
3dt[:,' .<
- 4- cr <0 ... h-
_ lo
- Itf
TE

7r. a+ F= i sinO+cosQ-cotg
cosec0-sec0*tan0

5 t2t2
I

_13 13 5
1313 5 f

5 t212
25+60-156
J?= 65 .'l16
sinp-
5
- 156 -65+25=-71"60
65

..6
sinp- -213
3
377
2J' tanQ -cotQ =a
sin B=
3
+ a2+4=tanz 6 + cot2 e-2tan0 . cote + 4
qt a2 +4 = tanag + cot2 e - z + 4
=
I
srno =, =a2+4=tan26+cot2g+2
+ a2 * 4=( 1 + tan2g ) + (r + cot2e )
BC = J?= + az t 4 =sec26 + cosec2e ......... (il
iTsTI b=cosg +sing
=2 J'
>b2=(cosg +sine)2
a+F=i
TC

=b2- | =cos2e + sin2g + 2sine . cose - 1

 +b2-1-1+2sin0.cose -1 Zg. cosec e,= Ji
+ G2- 1)2 = 4sin2g . cos2e ...............(ii)
qfrfiur (il aq (iil t, l_l
= Jz = sinA =O
Gz + +) $2- 1)2 = (sec2g + cosec2g ) . a sin2e . cos26 "irrA
slr[,

+ (a2+ 4) (b2- 1)2 = (#.#) . 4sin2e . cosze cosA= frffi

cosA =
+(a2++)62 -t)2= .n.,^rs.cos2s =
[#)
+(aa+a)G2-r)2=Lx4
-4
76. x2*y2=L
q54f5x=sinQ, y=cos 1

sinzg +cos20 =1 sinA @

= ' tanA=--- = 1 =1
3lt[; ry= sin Q . cos Q coSA
o
1
=r(;sine.cose) cotA=
tanA =
1

1 $[t[:, -
=;sin20
2sin2A+ 3cot2A
=f 4tan2A-.or2A
1

,'r ,"(#)' +sx(r)2

77. sinQ +cosQ =./2 sin(00"- g;

+sin$ +cosQ = J[ cosg
+ sin O = J, cosQ - cosQ
(.8 - t)
= sing =cosg
sinO cos0(.,8 - 1)
= .ote = .*e (z+zsino)(r-sine)
+tang =( Ji -t) ( 1 + cos0)(2
- 2cos0)
I 2(J+ sin0)(1- sinO)
:+cote =
rel - 2$+ cos0)(1- cosO)

+ cote = ml
t , (,..E +r) =
2(o-sinze)
@*1) 20- cos'O)

(.,8 + r) Ji +t .o.2 o (ts)2

.'.cote= = =.t2+l = "
=coto =
2_L 1 [e]
"_i,P,e
78. v{1 t, _ 225
tanQ *cot0 =2 64
+ (tan g + cot e)z - (ilz sl. F(qr t f+,
+tan?g +cot2g +2tan0.cot6 =4
b2
+tan2g +cot2Q+2=4 ^2
.'. tan2g +cotzO =2
?- v,
 a2 b2
_1+sinx_ (t+sinx)' =(1*sinx)'
=;%inh--tP*"Po 1-sinx (1+sinx)(1-sinx) 1-sinzx
11
-= ---------
sin2o tanzo _ (r * sin x)' _ (r * sin x)'
cos' , cosx
= cosec2e - cotze
-1 1+sinx
82. vST t, )'
=-:SeCx+tanX
cos x
m2+n2
= (a cose + b sine )2 + (a sing - b cose )2
89. "'PSlnx:93 sin, = I
* q
= (a2cos2e b2 sinzg + 2ab cos0 . sine ) + (a2 sinz6 + b2
cos2e -2ab sing . cos0 )
= a2(cos2 g + sinz 0 ) + bz (sinz g + cos2 0 )

- a2 +b2
sinQ +sin2e -1 Je'z n'z.tanx= ,77 'fi:q-q"
+sin0 =1-sin2e {P"
90. 4sinzx *4 cosz x = 4(sinzx * cos2 x\ = 4 x 7 = 4
= sine = cos20
ffi, 91. (r + tancr)(1+ tanB)
co.26 + co.4e =1 + tan cr + tan p + tan cr.tan B

= cos2e + (cos2e )2
'.'cr + ,4,
B=+= 45o .
6 + sinz e ... tan(cr* 9)= tan45o= 1
= cos2
=l tana+tanp _.,
cose +sinQ -.[ cosQ 1--L
1- tana.tan B
=+
(cos + sin 0 )2 = 2cos2 g
6 + tano + tan B = 1 - tancr.tanp
+ cos20 + sinzg + 2cos0. sine = 2cos2e
qn
{mfi'{ur (i) d tan cr + tan B tF'T rqi qr
+ cos2Q -2cos0 . sing = sinze
1+ 1- tano.tanp * tancr.tanp = 2
cos2Q - 2cos0 . sin g + sin20 = 2sinze
= 92. 0 = 45o Y{ sin0 * cosO t5'T qFT rffiIrT ddf tf
tfr T1q'sinzgfrgi q{l
+ (cos
0 - sin 0 )2 = 2sin2 6 .'. sin45o+cos45o=+* l= J,
,12 ,12
.'. cosQ -sinQ = f sing
.'.lsine+cos0l<.,8
sin0- 2sin30
85.
2cos30 - cosO
93. A+B=90o=B:90o-A
sin0(1- zsin2o) tanAtanB+ tanAcotB sinz B
= coso(2cos2e-t) sinAsecB

sin0{1 -2(o-cor2e)} tanAtan(90'-A)+ tanAcot(90'-A) sin2(SO'-,t )

cosO(2cos2e - 1) sinAsec(90'-A)
sinO(2cos2e - 1)
- cos0(2cos20-1)
tanAcotA+ tanAtanA cos2A

=tan0
86. sec 5A = cosec (4 + 36') fr + tr"'z A -1 = J.rrr'I = tanA
+ sec 5A = sec [90" - (4 + s6')l e4. EUq ABC d,
sec 5A = sec (s+" - A)
= A+B+C=180o
.'.5A=54"-A
64=54" (+n ir$F 2 t,ntr ti q{)
= A= =B+c=180o-A
.'. 9o
B+c
:90'-A
= 220
 (il + (ii)
=,ur['*t)=run[ro'-o)
\. 2 / ( 2) 2XZ:L4 .'- XZ = 7

( s+c\ e
.'.YZ=7 -2:5
= on[ , )= "o., .'. sec)( =-------
7
2,16
95. tan35o.tan45o.tan55o
: tan 35". 1. tan (90' -35")
5
tanX - -----
2J6
: tan 35o.1.cot35o= 1.1 = 1
7-* 5-
[t*( OO'-0) - cot 0; tan 0.cot0 = 1] .'.secX+tanx = 2J6 2J6
96. sec(70+28") 12 r;
- -={b
= cosec(30"-a"e) 2,,16

: tDO.Z=sin0+cos0
+ sec(7e + 28") sec(90o-(So'-se))
+ 28"= 90'-30"+30 = 22 = sin2 0 + cos2 0+ 2sinO.cos 0:1+ 2sinO.cos0
=70 .''0<e<90'
40 = 90o-30" -28"=32"
= .'.sin0<1;cos0<1
=o=32o-8o
4
...22 <2

97. tan 20.tan 30 = 1

...2 < J'
1 101. sin20+cos20+sec20+cosec2e+tur20+
"ofe
=tan30- tan20 =cot20 : 2(tan20+ cot2 0)
1 + sec2 e - tane 0 +cosec20 -cot2 0+
\/
+ tan30 = tan(90"-20)
= 3+ 2(tan0 -cot 0)2 +2)>7
50=90o
=30:90'-20 =
.'. 0 = 18o Hitd. (tan0 - cot0)3 > 0
{ 102. cot a *tand =m
98. sin17o=
v = 1+tanza =mtana (i)
= sec2 d, = mtano
cos17o= d-.trf 1'r"
.1
' cosa - cos (t = l'r
1

= secd - secd
+ secza - -=n
| =n secd
tanz a =nsecd
=
+ tana a = razsec2 d,
sin730= sin(90"-t7')
> tana a = n2m tana (qfrs{ur(il t)
: cos17o 3 tans a = n2m
.'. sec17o-sin73' 1

+ tan, = (r2..,)B
\/
v lT *2
-v2 -v2
+,2
Jf -7 v vJf -.' gel secz d, = m ("'*),
1

99. XZ-YZ = 2.....(i)

.'. sec2a -tanza =L
+ru2KYz +YZ2 11
2,16 = *(r'2-)' - (-"')' =,
=(zJe)' =xZ2 -YZz
11
+ 24 = (xz -yz)(xz +yz) .'.-(-"2)B -n(*"2)B = 1

+ XZ +YZ:12....(ii)
 1-cosa+sina l-cosa+sina l-cosa+sina sin3g cos2 g + cos3 o.sin2 o
io3' l+sina = 1*rirro ^ l+cosd+sind sinO *cos0

(t+sina)2 -"or2 o sin2g.cos2 9(sin0 + cos0)

= (, * sina)(l+ cosd +sina)
' = sing+cosd
= sin2 6 . cosz 6
l+2sina+sin2 a+sin2 a
(1+ sin a)(t + cosd + sina)
108. cot 0 I +sin0 ) = 4m .......(i)
cot 0 (1 - sin 0) = 4n .......(ii)
2srna(l + sina) (i) x (ii)
= (t *sina)(1 + cosa +sina) cot2g (1 -sin'A) 16mn
2sina
cotzg.coszg =16mn
= l--------------- = v
I+cosd+slna aor4e
- .---d= =
srn'U
16mn .........(iii)
sin2v 1* .cosy sin y
104.1-- I+cosY+ Srny -. r-cosy fqq .rq {-qtl 6i ert or qzri q{,
cotz g [(t + sin e)2 - (t - sin 0)') = 16 (m2 - n')
1f cosy-sin2 y 1+cos2y - sin2y cotz g . 4sin 0 = L6(m'- n')
T
-T sinY(1-cosY)
1*cosy
cotL6.sinB =4(m2-n2)
cos y + cos2y Sqt Ff +ri q{
I+cosy + cotaB . sinz 0 =76 (m2-n2)2
bc
105. sin 0 fcose =;F sino.cose=; --.--T
"o"4e- = 16 (mz - n')'
= srn-d
... (m2 -n\, =mn 1ffi. (iiil tl
.'. (sinp +cos e)'=*a"
109. r\
I'rl
b"
+ sin2g *cos2 g +2sine.cose o I 1
I\. ,^* r . r^isinz1.cosze
a' - -cos'6 sin"O
cos"0 -srn'dl)
2c b2
+1+-=---d
aa' ( sin2e )
.a2+2ac-b2=0
I-+-
"o"ze 'l-sinagJI sm-o
=[r-"or4e
. o
'cos'o
o

sinSa .o.34 l-2sin20."or2e

106. ------=-.coszp" * o .sinzB -
cos'0 sin'0 sin0.cosd

sin3d "o.30
1-
:r-2sin2o."or2e
cosg ' sind sin0.cos0
sinz6."orr, =
,*"o3. .#k
sinag +.or4 o I -2sin2o."o"2e
sing.cos0 sin0.cos0
.o.ae(t + sin2o)+ sinao(r
\/\./ + cos2o)

(sin2a + e)2 - 2sin2e.cos2 o L-2sin20."o"2e (r + +.o'2e)

"o"2
sin0.cos0
-0 'i"2e)(r
sin0.cos0
.or40 + cos40.sin2e + sinae + sin40.cos20

1^, Tb-Tu _ sin30+cos3g-(si"sa+.or5a) = (r + rirr2o + cos2o + sin2o.cos2o)

'Ii sin0 * cos0
.or4e + sina0 + sin20 .o.2e(.o.2e + sin2a)
sin3o - sin5o + cosSo -.o.50 2+ sin20.cos20
sin0 * cos0

sin2a) + cos3o (r (.o.2e +.i.r20)2 - 2cos20.sin20 + sin2o.cos2e

sin3e (r - -.o.24) -
sinO * cosO 2+ sin20.cos20
 I - sin20cos20 = cosec2 0 -cotz 0 =l
= (cosecA +coto)(cosecQ
2+ sin20..o.2e = -cote)=t
110. sec 0 +tarr? =2 ...... .. (i)
cosecg +cotQ
=
.'. secz Q -tan20 =L
'12
(ii)
cosecO-cotO 7
=+ (sec B + tan 0) Gec g -tan 0)=L
(il + (ii)
1
+ e -tane =,
sec .........(ii)
72 49+ 4 53
q++.{ur (il +(iil t,
2cosecg = i*i = t4 L4

... sec0 +tarrg *secQ -tanQ Q

.'. cosec, = 28
_zlt
15
-Or -2_- 2 113. ... nrad = 180o
33 I 80'
+ 2 sece =12 .'. sec 0
5 ;rad=;
DC
x
lf
4
Lll.. @. m\ (P + m2 + 3) 37 7x3x180' 378'=
=+ 0 -sine)z
(cosec =-x-
522 x 180"=
11 11 l,rna)
[ 11/
(secB -cos0)z
{(cosecg - sin 0)z + (secA - cos 0)z + 3} =Bn'(#,60') =rn'(T) =rn"(rr*) =B4o2L(*.uo')
' -sin o\'(
=((sin0 ' -.o"e)')
/ [cos0 = 3402L',(#) = 3402'',4e"

- sin r)' * (* - cos')' .,i 114. 10 rad= (57"16'21") x 10

{t* = 570o + 160' + 210" = 570o + 2" 40'+ 3'30" = 572o43'30"
115. 5E6I{ = (29"46'55") x 5
(t-.i.r'
_l_tt_t e\2 ( t-.o.' a)' = L45"230'275" = 145" + 3"50' + 4'35" = 148o54'35"
-[ sino .l I coso ) 116. 100 {S'f$ = (15'49'50") x 100
= 1500.4900'5000" = 1500o + (91" 40') + g3'20"
qsoo 4o booo
( 60 = ,, 60' 60 = ss?g')
( ...
60)
{t=*)'.[';::,')'.,] = 1500o + 81o40' + 1"23'20"
= L582" 63'20" = 1583o3'20"

qfi frTe {
=
[*)' [*)' {[*)'.[*)'.,]
112. q[6qr
# - B !qq;1 q'nhnr
... ldqq;1 =360o
o sina g ["ora o -' sin2e ,-' o'l .'. 3+FF{ = 1080o
= "or4 *
;,hE ;;%. i '1"%- ;;'e "l
118. qf6qr 1ffis { #=; ffi{ e,rmr ir
=cos2g xsin2r ... 1*FF-t =2rrad
{ }
'+
= cos6 B + sin6 B 3cos2 g + sinz 0
L2n
+sin'd.)} b *FF{= U rad
= {(cos'B + sin2 e)' -3 cos20 . sin2g . (cos2B
*3coszo 'sinze 11e. d,
^OAB
[... ar+bB=(a+b)B 15Lx
sinO- BO=, +0=30'=6
- 1 - 3 cosz6 . sinz e + B
"or, e11[.='i' [...oo = oB = Bo G)]

112. cosec Q -cott =f,........... (i) lud t BB'frqt Bo+fi. + S +q qt 2e =2"

t= f et'r
qrnr tr
 /./tl 22xG 132 6 \^
- -:El
3;kl: e
r'" 8-30 +l=l0r =- 7 - 7
- =r8;Ei
--7
lox22 220 \A 3 \^ 7f
= 7 = 7 {tfl-317I[rffl 126.90o=;z

/10 1 I
e=; = l= 0. ,= Zx roofia
3.r4
=; x 100 = 15T friR

1zz. gq6 6qqfi{ 30fr{e t s{q m'S sqt sofrsqg +l qqq

rq qt f{ Et 61 dA q$ r Tcr {sodmT +tur eq m,rfi
=(ff) =('+) =5'(*'uo') =r"[#) frr

tr gsfttq qE 8i q 30'
"r; =25boq qtur rq
qa:frr
".
= s. (+s1) = u.nr, (+'oo") =s.+s,88,, ErrT$rrr tz8. qFrT tfi lvffir iqr*. E1 do t r tfr E1 $ qt t'
i.e. OC = 1ffi
LZt. e=OO'= f iTsTI /=5gffi
AocFt, zco*= l ,ro,= ,u,=
[l)
sin (zCOF) = OC=
.. (i)
150
t
I 150 \^ m 150x7
'e
r=- = lt
5 tt7 = 47.72 ffi t(... qrq =1ffi; cF=i *,
L22.
11
VM=30H t ArFI wtq o-ri qr, . (r)'
+ srn
1

lz)= 2t
11
+ A=A=11x15-30=135o= 180'
135 W, pqg( dJr d, d sin e = e
7x30=30x1 (qn r€ T6 radian + tdq rs +)
3tt
r rad=7 ,ud
(l =+ = (i'#) ,ud=*
12s. qtrqr 1+6rs {# qR Wr qror i' (zrtgo\ 90 \^ 9ox7 6 \
=> I=
l;)=':tfi, ;cm=1411 tfr
qf6qr ltq.'s { # x2x tfsr{ af1 q..q61 }1 Lzs. zlq.ci { yd effii 3rql + qrt *{ Trfr * = 860'
ef,fl: tntut d rro ifsfi Tttr qti { qf6qr
l qozr d vr* srqi s{H + qd' sfu Trrfi + = 9# = lbo
110
il 110x10x7 \ +q") { fd erqi orq + qrd sTt{ Trfr * = lb" x 4 =60o
-' = 7'3 t'flrg qrq
Z+ em
x2x = ffi
... 60 fu{c t
102 td emi erq + qrii S{ Trfr t= rs'
/183 12fr{e { g,d ffi d q-ir' st{ {q(i EEffi
124. e =;
= e = 60 =, rad = 0.3 rad =r"
31qr

..- lrad = 57" 16" 22'tffi$trT + qut 12 Fr=ra { ged ffi slql + qrd .frr {rfr *
... 0.Brad = 5.7" x 3 = 17't sTRI-fr dsII 18't eq - 60o+3o=63o

20" x 7t
180. tan45o = 1 de{r {s+ qrq 216 rTFI 1t *or tr
vs-r
125.20"=1g0p=;rad 131. ffiq sinx= cos y$gr t+ x rTsl v{ =p+trr t, re
lt
I tt 22 \A x=y- x+y= L
e=; - l=0 t= g, S+= 7xS x 54{[rTI 2
eicT: fq+w (.) vfr *r

*-***-"*"**s
A NCED IE MATIC S
 lt\
132.0<x<;S'f,dq,
z
(ii)t (iv)

cos2x dell sinzx, 0 dsTI 1 * fiq { +dr t't era , t"rse (d gfr
tr (iii)dqr (iv) t sina*cosd = :q (v)
133. p = tantzx t cotzx= (tan, - cotr)z t 2tanx cotx
=(tar1;6-cotx)2 +2 > 0+2=2
Yf{ (sin a *cos a)' = sin2 a *cos2 a *2 sin d,
.cos
d,

13a. $ur lm aq qrtfr tm sinx, t of

go'it-.fi-qGiTT (increasing)
( p)' 2 p2 q,+2
r =l =1+ q,
+'\q) - = --r= ..p2= qz +2q=q(q+2)
tr q' q
.'. sin y > sin x -
135.0<e<Q<gO"+rdq l42. secQ -tanl -
0<sin2g,cosze <l
... sinzg +coszg<2

186. 6rr vri t errnro t .lo,i) t sec,p, 1t - qaor tt

L43. x-h - asecl cosO = *
.'.p>1
1Bz. f(qr +, b
y-k=bcosecB
cosg +
= sing = y_k
.f sin 0 =2
'.'coszB +sinz0 =L
j.o. o .
+sino - 1 ( a )2 ( u )'
.[r-o.,J -1
= 0 + cos3Oo sin d -
sin3Oo cos 1 l*r)
+sin(30"+ 0)=sin90' 144. sinA- .6 cosA= /f cosA6i qri q-(i qt,
+30o* d =90o sin2A + 6cos2A- 2 J6 sinA cosA = 7cos2A
lf
+ sin2A = coszA + 2 J6 sinA cosA
"'0-60'=3 qti ilrtr 6 sinzA# qt,
138. AOAB d,
7sin2A = coszA + 2 J6 sinA cosA + 6 sinzA
AB
cos4Oo = OB + ZsinzA = (cosA + ,16 sinA)2

r ... cosA+ uf, sinA= t fi sinA

:;, cos 40" = R
.'. r=Rcos4Oo
145. Til sT +rl, sin B + cos 6 = -!
d 5 +1 frqt R cos +o'frfrt
o{ct: 40'S 3Iq{I{T
189. qR 4r 0z dtr q d 0z yerq qgqf{r {d, ir6r singr il ad s.r {uI{t6e, sin g cos o=9
EFI sin erd qH t qsJ +nr rqt+. cosgr trl qH cos2z '.' (sinp +cos g)'=sinzB *"o"lr+,2sin e.cose
t *cr tor tr , .12
I-b\ c
l- | =L*2- a
11 * Bsinzo +2cos2o sin2e +2 \a/
140. P- 7sinz e icoszB = + a2 -bz * 2ac= 0
'..0 s sin2B < 1 + a2 + 2ac=b2
31 qfi' 6q5, c'*5i R, (a + c)'= b2 + c2
=1,P=e-, 146; cos, U1 3ftrqrq qH 1*
141. {d S'f +rf = sec a r cosec d = p ... sin (cos x) E-t erftrfr-oq HFII = sinl
......(il 147. (f,df,ch. sin26 *cos2 g+tanzgtcotz0 *secz0 +cosecze
ad t5T {urTs.d =secd cosec a =q. (iil = (sinzg *cosz g)+tanz g +cotz e +O rtanz (f + cot2 e)
A)
. I + --:-
I sina+cosa
------- = 1 | 1 g + cotz 0)
+ 1 + 2 (tanz
tr.l tr.' (iii)
cosa sma =D = aorrarro =P
=3+2{(tanQ _cotg)z+2}
 YQ (tan 0 -cote)' >O + sin2x. sirr'y - siny sin2.r sin2y sirx. siny
1 2 sin-r. - - - 2
5aq5 > 3.+z(o+z) =(it'q5'17 aoa2r.aoa2y
148.cos2a lcos2fl =2
art ejrq t, qq cosz a r[eTT cos2 B +n' H't rTFT 1 6], ffi6 1+ sin2x.sin2y -sin2x - sin2y (r -.i"'r)(r - 'i"2v)
qd cos'a ilstrT cosz B EH { t ffi H'r fi rffiT lt qg-r rfr =r, cos-x.cos-y aor2r.aoa2y

*cr tr
cos2p-L _ aor2r..o.2y
O _.
= ct=F =0"
cos2d, =ldSrlT I
cos-.r.cos-y -r
SikT: tan3 a + sins g
= (tan0")3 + (sin0")5 = 0 150. "or2o
149. (secx. secy *tanx. tany)'-(secx. tany*tanx. secy)2
' cot"0 -coso?
-6-------------_-_d

( cos20 =3cot2g-3cos26
t sin )2 ( l.rirr u
r.sin v sin x )2 =
-t't-t'lcos x.cos y cos.r.cos
\ / i \ cos .r.cos y cos r.cos y / +
3cos2o 3
4cos' 0 =---. ,) ^ + sur A
,lZ .'. 0 = 60"
srn-d
( 1 + sin ,..i., 'r, \2 ( sin x+ sin v \2
- (. cos x.cos y .l - [ .o. r..o, y .J