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  4. The value of cot -1((√1+ sin x+√1- sin x/√1+ sin x-√1- sin x)) is equal to

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Q. The value of $\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$ is equal to

ManipalManipal 2020

Solution:

Let $I=\cot ^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$
$=\cot ^{-1}\left[\frac{\sqrt{\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}}+\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2}}}{\left.\sqrt{\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}}-\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^{2}}\right]}\right.$
$=\cot ^{-1}\left(\frac{\sin \frac{x}{2}+\cos \frac{x}{2}+\cos \frac{x}{2}-\sin \frac{x}{2}}{\sin \frac{x}{2}+\cos \frac{x}{2}-\cos \frac{x}{2}+\sin \frac{x}{2}}\right)$
$=\cot ^{-1}\left(\frac{2 \cos \frac{x}{2}}{2 \sin \frac{x}{2}}\right)=\cot ^{-1}\left(\cot \frac{x}{2}\right)=\frac{x}{2}$