Differentiate ^ -1 ( 1+ ) with respect to x. - Vidyadip

Question

Differentiate $\tan^{-1}\Big(\frac{1+\cos\text{x}}{\sin\text{x}}\Big)$ with respect to x.

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Answer

Let $\tan^{-1}\Big(\frac{1+\cos\text{x}}{\sin\text{x}}\Big)=\text{y}$
$\text{y}=\tan^{-1}\bigg(\frac{2\cos^2\frac{\text{x}}{2}}{2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}\bigg)$
$=\tan^{-1}\Big(\cot\frac{\text{x}}{2}\Big)$
$=\tan^{-1}\Big[\tan\Big(\frac{\pi}{2}-\frac{\text{x}}{2}\Big)\Big]$
$\text{y}=\frac{\pi}{2}-\frac{\text{x}}{2}$
$\frac{\text{dy}}{\text{dx}}=0-\frac{1}{2}=\frac{-1}{2}$

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