Differentiate the following functions with respect to x :

Question:

Differentiate the following functions with respect to $x$ :

$\tan ^{-1}\left\{\frac{\cos x+\sin x}{\cos x-\sin x}\right\},-\frac{\pi}{4}

Solution:

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

Dividing numerator and denominator by $\cos x$

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

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

$y=\tan ^{-1}\left(\frac{\tan \left(\frac{\pi}{4}\right)+\tan x}{1-\tan \left(\frac{\pi}{4}\right) \tan x}\right)$

Using, $\tan (x+y)=\left(\frac{\tan x+\tan y}{1-\tan x \tan y}\right)$

$y=\tan ^{-1}\left(\tan \left(\frac{\pi}{4}+x\right)\right)$

$y=\frac{\pi}{4}+x$

Differentiating w.r.t $x$ we get

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\pi}{4}+\mathrm{x}\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}}=0+1$

$\frac{\mathrm{dy}}{\mathrm{dx}}=1$

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