${\cot }^{-1}\left(\frac{\sqrt{1-\sin \mathrm{x}} +\sqrt{1+\sin \mathrm{x}}}{\sqrt{1-\sin \mathrm{x}}-\sqrt{1+\sin \mathrm{x}}}\right)=$

$$\begin{gathered} 1+\sin \mathrm{x}={\cos }^2 \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+{\sin }^2 \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2+2\sin $}\right.\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \cos \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ ={\left(\cos \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\sin \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^2 \\ {\cot }^{-1}\left[\frac{\cos \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\sin \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\cos \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+\sin \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\cos \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\sin \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\cos \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\sin \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right] \\ {\cot }^{-1}\left[\frac{2\cos \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{-2\sin \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]={\cot }^{-1}\left[-\cot \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right]=\mathrm{\pi }-\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \end{gathered}$$