Question

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

Answer 1

Let $y={\tan }^{-1}\left(\frac{1+\cos x}{\sin x}\right)$

$={\tan }^{-1}\left(\frac{2{\cos }^2\frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}}\right)$

$={\tan }^{-1}\left(\cot \frac{x}{2}\right)$

$={\tan }^{-1}\left(\tan (\frac{\pi }{2}-\frac{x}{2})\right)$

$y=\frac{\pi }{2}-\frac{x}{2}$

Differentiating both sides w.r.t. $x$, we have

$\frac{dy}{dx}=0-\frac{1}{2}=-\frac{1}{2}$