The value of ${\cot }^{-1}\left[\frac{\sqrt{1-\sin \mathrm{x}}+\sqrt{1+\sin \mathrm{x}}}{\sqrt{1-\sin \mathrm{x}}-\sqrt{1+\sin \mathrm{x}}}\right]$ is

Solution:

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