frac{d}{d x} tan ^{-1}left[frac{sqrt{1+sin x} | vedclass

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(C) Let $y = 	an ^{-1}\left[\frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{\sqrt{1+\sin x}+\sqrt{1-\sin x}}ight]$.We know that $1 + \sin x = \cos^2 \frac{x

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$\frac{1}{2}$

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(C) Let $y = 	an ^{-1}\left[\frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{\sqrt{1+\sin x}+\sqrt{1-\sin x}}ight]$.We know that $1 + \sin x = \cos^2 \frac{x}{2} + \sin^2 \frac{x}{2} + 2 \sin \frac{x}{2} \cos \frac{x}{2} = (\cos \frac{x}{2} + \sin \frac{x}{2})^2$.Similarly,$1 - \sin x = (\cos \frac{x}{2} - \sin \frac{x}{2})^2$.Substituting these into the expression:$y = 	an ^{-1}\left[\frac{(\cos \frac{x}{2} + \sin \frac{x}{2}) - (\cos \frac{x}{2} - \sin \frac{x}{2})}{(\cos \frac{x}{2} + \sin \frac{x}{2}) + (\cos \frac{x}{2} - \sin \frac{x}{2})}ight]$$y = 	an ^{-1}\left[\frac{2 \sin \frac{x}{2}}{2 \cos \frac{x}{2}}ight] = 	an ^{-1}(	an \frac{x}{2}) = \frac{x}{2}$.Now,differentiating with respect to $x$:$\frac{dy}{dx} = \frac{d}{dx}(\frac{x}{2}) = \frac{1}{2}$.

$\frac{d}{d x} \tan ^{-1}\left[\frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{\sqrt{1+\sin x}+\sqrt{1-\sin x}}\right]$ is equal to

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