$2{\tan }^{-1}\left(\tan \frac{\mathrm{\theta }}{2}\tan \frac{\mathrm{\varphi }}{2}\right)=$

We have, $2{\tan }^{-1}\left(\tan \frac{\mathrm{\theta }}{2}\tan \frac{\mathrm{\varphi }}{2}\right)$

$={\cos }^{-1}\left[\frac{1-{\tan }^2\frac{\mathrm{\theta }}{2}{\tan }^2\frac{\mathrm{\varphi }}{2}}{1+{\tan }^2\frac{\mathrm{\theta }}{2}{\tan }^2\frac{\mathrm{\varphi }}{2}}\right]={\cos }^{-1}\left[\frac{1-\frac{{\sin }^2\frac{\mathrm{\theta }}{2}{\sin }^2\frac{\mathrm{\varphi }}{2}}{{\cos }^2\frac{\mathrm{\theta }}{2}{\cos }^2\frac{\mathrm{\varphi }}{2}}}{1+\frac{{\sin }^2\frac{\mathrm{\theta }}{2}{\sin }^2\frac{\mathrm{\varphi }}{2}}{{\cos }^2\frac{\mathrm{\theta }}{2}{\cos }^2\frac{\mathrm{\varphi }}{2}}}\right]$

$\begin{array}{l}={\cos }^{-1}\left[\frac{{\cos }^2\frac{\mathrm{\theta }}{2}{\cos }^2\frac{\mathrm{\varphi }}{2}-{\sin }^2\frac{\mathrm{\theta }}{2}{\sin }^2\frac{\mathrm{\varphi }}{2}}{{\cos }^2\frac{\mathrm{\theta }}{2}{\cos }^2\frac{\mathrm{\varphi }}{2}+{\sin }^2\frac{\mathrm{\theta }}{2}{\sin }^2\frac{\mathrm{\varphi }}{2}}\right]\\ ={\cos }^{-1}\left[\frac{\left(\cos \frac{\mathrm{\theta }}{2}\cos \frac{\mathrm{\varphi }}{2}+\sin \frac{\mathrm{\theta }}{2}\sin \frac{\mathrm{\varphi }}{2}\right)\left(\cos \frac{\mathrm{\theta }}{2}\cos \frac{\mathrm{\varphi }}{2}-\sin \frac{\mathrm{\theta }}{2}\sin \frac{\mathrm{\varphi }}{2}\right)}{{\left(\cos \frac{\mathrm{\theta }}{2}\cos \frac{\mathrm{\varphi }}{2}+\sin \frac{\mathrm{\theta }}{2}\sin \frac{\mathrm{\varphi }}{2}\right)}^2-2\sin \frac{\mathrm{\theta }}{2}\cos \frac{\mathrm{\theta }}{2}\sin \frac{\mathrm{\varphi }}{2}\cos \frac{\mathrm{\varphi }}{2}}\right]\\ \begin{array}{l}={\cos }^{-1}\left[\frac{\cos \left(\frac{\mathrm{\theta }-\mathrm{\varphi }}{2}\right)\cos \left(\frac{\mathrm{\theta }+\mathrm{\varphi }}{2}\right)}{{\cos }^2\frac{\mathrm{\theta }-\mathrm{\varphi }}{2}-\frac{\sin \mathrm{\theta sin}\mathrm{\varphi }}{2}}\right]\\ ={\cos }^{-1}\left[\frac{2\cos \left(\frac{\mathrm{\theta }-\mathrm{\varphi }}{2}\right)\cos \left(\frac{\mathrm{\theta }+\mathrm{\varphi }}{2}\right)}{2{\cos }^2\frac{\mathrm{\theta }-\mathrm{\varphi }}{2}-\sin \mathrm{\theta sin}\mathrm{\varphi }}\right]\\ ={\cos }^{-1}\left[\frac{\cos \mathrm{\theta }+\cos \mathrm{\varphi }}{1+\cos (\mathrm{\theta }-\mathrm{\varphi })-\sin \mathrm{\theta sin}\mathrm{\varphi }}\right]\end{array}\\ \begin{array}{l}={\cos }^{-1}\left[\frac{\cos \mathrm{\theta }+\cos \mathrm{\varphi }}{1+\cos \mathrm{\theta cos}\mathrm{\varphi }+\sin \mathrm{\theta sin}\mathrm{\varphi }-\sin \mathrm{\theta sin}\mathrm{\varphi }}\right]\\ ={\cos }^{-1}\left[\frac{\cos \mathrm{\theta }+\cos \mathrm{\varphi }}{1+\cos \mathrm{\theta cos}\mathrm{\varphi }}\right]\end{array}\end{array}$