Question

Differentiate ${\tan }^{-1}\left(\frac{\sin x}{1+\cos x}\right)$ w.r.t. $x$

Answer 1

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

$={\tan }^{-1}\left[\frac{\frac{2\tan x}{1+{\tan }^{2}x}}{1+\frac{1-{\tan }^{2}x}{1+{\tan }^{2}x}}\right]$

$={\tan }^{-1}\left[\frac{\frac{2\tan x}{1+{\tan }^{2}x}}{\frac{1+{\tan }^{2}x1-{\tan }^{2}x}{1+{\tan }^{2}x}}\right]$

$={\tan }^{-1}\left[\frac{2\tan x}{2}\right]$

$Z={\tan }^{-1}\left(\tan x\right)=x$

Now,

$\frac{d}{dx}\left(Z\right)=\frac{d}{dx}\left(x\right)=1$