Question

Write the function in the simplest form :

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

Answer 1

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

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

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

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

$=\frac{\pi }{4}-x$