Which of these angles is a value of $\tan^{-1} x$ for some real $x$?

Explanation

1. 01

   $\tan^{-1} x$ takes exactly the values in the open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

2. 02

   Only $\frac{2\pi}{5}$ is inside, since $\frac{2\pi}{5} < \frac{\pi}{2}$: it is $\tan^{-1}\left(\tan \frac{2\pi}{5}\right)$.