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