Write the following function in the simplest form: ^ -1 ( x- x x+… - Vidyadip

Question

Write the following function in the simplest form:
$\tan^{-1}\bigg(\frac{\cos x-\sin x}{\cos x+\sin x}\bigg), x<{\pi}$

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Answer

$\tan^{-1}\bigg(\frac{\cos x-\sin x}{{\cos x+\sin x}}\bigg)$
Dividing the numerator and denominator by cos x,
$=\tan^{-1}\bigg(\frac{1-\tan x}{1+\tan x}\bigg)=\tan^{-1}\bigg(\frac{\tan\frac{\pi}{4}-\tan x}{1+\tan\frac{\pi}{4}\tan x}\bigg)$
$=\tan^{-1}\tan\bigg(\frac{\pi}{4}-x\bigg)=\frac{\pi}{4}-x$

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