relationship on Jacobi elliptic functions

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We know that if $\alpha+\beta+\gamma=\pi$, then $\cos{\alpha}^2+\cos{\beta}^2+\cos{\gamma}^2+2\cos{\alpha}\cos{\beta}\cos{\gamma}=1$.

My question is that if $\alpha+\beta+\gamma=\pi$, is there any relationship between $cn(\alpha,k)$?