Differentiate the following with respect to x | vedclass

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(B) Let $f(x) = 	an ^{-1}\left(\frac{\sin x}{1+\cos x}ight)$.Using trigonometric identities $\sin x = 2 \sin \left(\frac{x}{2}ight) \cos \left(\f

Vedclass · Accepted Answer

$\frac{1}{2}$

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(B) Let $f(x) = 	an ^{-1}\left(\frac{\sin x}{1+\cos x}ight)$.Using trigonometric identities $\sin x = 2 \sin \left(\frac{x}{2}ight) \cos \left(\frac{x}{2}ight)$ and $1 + \cos x = 2 \cos ^{2} \left(\frac{x}{2}ight)$,we get:$f(x) = 	an ^{-1}\left[\frac{2 \sin \left(\frac{x}{2}ight) \cos \left(\frac{x}{2}ight)}{2 \cos ^{2} \left(\frac{x}{2}ight)}ight]$$f(x) = 	an ^{-1}\left[	an \left(\frac{x}{2}ight)ight]$$f(x) = \frac{x}{2}$Now,differentiating with respect to $x$:$f'(x) = \frac{d}{dx}\left(\frac{x}{2}ight) = \frac{1}{2}$.

Differentiate the following with respect to $x$: $\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right)$

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