Question:
If $y(x)=\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right), x \in\left(\frac{\pi}{2}, \pi\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\frac{5 \pi}{6}$ is:
Correct Option: 1
Solution:
$y(x)=\cot ^{-1}\left[\frac{\cos \frac{x}{2}+\sin \frac{x}{2}+\sin \frac{x}{2}-\cos \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}-\sin \frac{x}{2}+\cos \frac{x}{2}}\right]$
$y(x)=\cot ^{-1}\left(\tan \frac{x}{2}\right)=\frac{\pi}{2}-\frac{x}{2}$
$y^{\prime}(x)=\frac{-1}{2}$