Question

Write the function in the simplest form:

Answer 1

$$\begin{gathered} {\tan }^{-1}\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right) \\ ={\tan }^{-1}\left(\frac{1-{\displaystyle \frac{\sin x}{\cos x}}}{1+{\displaystyle \frac{\sin x}{\cos x}}}\right) \\ ={\tan }^{-1}\left(\frac{1-\tan x}{1+\tan x}\right) \\ ={\tan }^{-1}\left(1\right)-{\tan }^{-1}\left(\tan x\right)\left({\tan }^{-1}\frac{x-y}{1+xy}={\tan }^{-1}x-{\tan }^{-1}y\right) \\ =\frac{\mathrm{\pi }}{4}-x \end{gathered}$$