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Q. 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 $\frac{ dy }{ dx }$ at $x =\frac{5 \pi}{6}$ is:

JEE MainJEE Main 2021Inverse Trigonometric Functions

Solution:

$y(x)=\cot ^{-1}\left[\frac{\cos \frac{x}{2}+\sin \frac{x}{2}+\sin \frac{x}{2}-\cos \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}-\sin \frac{x}{2}+\cos \frac{x}{2}}\right]$
$y(x)=\cot ^{-1}\left(\tan \frac{x}{2}\right)=\frac{\pi}{2}-\frac{x}{2}$
$y^{\prime}(x)=\frac{-1}{2}$