What is the value of $\cos(\tan^{-1}(\tan 2))$?

Question

What is the value of $\cos(\tan^{-1}(\tan 2))$?

Am I thinking correct? $\tan 2$ is negative so $\tan^{-1}$ and $-\tan 2$ cancel each other giving $\cos(-2)$ which finally gives the answer as $-\cos 2$? Am I correct? Please explain if not.

Answer 1

$\arctan \tan x\equiv x\mod\pi$ and by definition, $\arctan t\in(-\frac\pi2,\frac\pi2)$. As $\;\frac\pi 2<2<\pi$, this means $\arctan \tan 2=2-\pi$, whence $$\cos(\arctan \tan 2)=\cos(2-\pi)=\color{red}{-\cos 2}.$$