Question

Write the function in the simplest form: ${\tan }^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right),0<x<\pi$

Answer 1

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

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

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

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