Addition formulas for Jacobi amplitude function

One can use addition theorems for Jacobi elliptic functions. In particular, since \begin{align} \sin(\mathrm{am}\;u)&=\mathrm{sn}\,u,\\ \cos(\mathrm{am}\;u)&=\mathrm{cn}\,u,\\ \sqrt{1-k^2\sin^2(\mathrm{am}\;u)}&=\mathrm{dn}\,u, \end{align} and, say, \begin{align} \mathrm{sn}(u+v)=\frac{\mathrm{sn}\,u\;\mathrm{cn}\,v\;\mathrm{dn}\,v+ \mathrm{sn}\,v\;\mathrm{cn}\,u\;\mathrm{dn}\,u}{1-k^2\mathrm{sn}^2u\;\mathrm{sn}^2v}, \end{align} one finds \begin{align} &\qquad \mathrm{am}(u+v)=\\ &=\arcsin\left(\frac{\sin(\mathrm{am}\;u)\cos(\mathrm{am}\;v) \sqrt{1-k^2\sin^2(\mathrm{am}\;v)}+ \sin(\mathrm{am}\;v)\cos(\mathrm{am}\;u) \sqrt{1-k^2\sin^2(\mathrm{am}\;u)}}{1-k^2\sin^2(\mathrm{am}\;u)\sin^2(\mathrm{am}\;v)}\right). \end{align}