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