frac{d}{dx}left( tan^{-1}sqrt{frac{1 + cos(x/ | vedclass

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(A) Let $y = 	an^{-1}\sqrt{\frac{1 + \cos(x/2)}{1 - \cos(x/2)}}$.Using the trigonometric identities $1 + \cos 	heta = 2\cos^2(	heta/2)$ and $1 - \c

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$- \frac{1}{4}$

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(A) Let $y = 	an^{-1}\sqrt{\frac{1 + \cos(x/2)}{1 - \cos(x/2)}}$.Using the trigonometric identities $1 + \cos 	heta = 2\cos^2(	heta/2)$ and $1 - \cos 	heta = 2\sin^2(	heta/2)$,we get:$y = 	an^{-1}\sqrt{\frac{2\cos^2(x/4)}{2\sin^2(x/4)}}$$y = 	an^{-1}\sqrt{\cot^2(x/4)}$$y = 	an^{-1}(\cot(x/4))$Since $\cot(x/4) = 	an(\frac{\pi}{2} - \frac{x}{4})$,we have:$y = 	an^{-1}(	an(\frac{\pi}{2} - \frac{x}{4})) = \frac{\pi}{2} - \frac{x}{4}$Differentiating with respect to $x$:$\frac{dy}{dx} = \frac{d}{dx}(\frac{\pi}{2} - \frac{x}{4}) = 0 - \frac{1}{4} = - \frac{1}{4}$.

$\frac{d}{dx}\left( \tan^{-1}\sqrt{\frac{1 + \cos(x/2)}{1 - \cos(x/2)}} \right)$ is equal to

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