\[\operatorname{cs}\left(2z,k\right)=\frac{{\operatorname{cs}}^{4}\left(z,k%
\right)-{k^{\prime}}^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}%
\left(z,k\right)\operatorname{ns}\left(z,k\right)}.\]