\[{\operatorname{gd}^{-1}}\left(x\right)=\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}%
\pi\right)=\ln\left(\sec x+\tan x\right)=\operatorname{arcsinh}\left(\tan x%
\right)=\operatorname{arccsch}\left(\cot x\right)=\operatorname{arccosh}\left(%
\sec x\right)=\operatorname{arcsech}\left(\cos x\right)=\operatorname{arctanh}%
\left(\sin x\right)=\operatorname{arccoth}\left(\csc x\right).\]