\[\operatorname{cn}\left(2z,k\right)=\frac{{\operatorname{cn}}^{2}\left(z,k%
\right)-{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,%
k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{\operatorname%
{cn}}^{4}\left(z,k\right)-{k^{\prime}}^{2}{\operatorname{sn}}^{4}\left(z,k%
\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)},\]