Find the values of

$\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$

We know that $\tan ^{-1}(\tan x)=x$ if $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, which is the principal value branch of $\tan ^{-1} x$.

Here, $\frac{3 \pi}{4} \notin\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$

Now, $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$ can be written as:

$\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)=\tan ^{-1}\left[-\tan \left(\frac{-3 \pi}{4}\right)\right]=\tan ^{-1}\left[-\tan \left(\pi-\frac{\pi}{4}\right)\right]$

$=\tan ^{-1}\left[-\tan \frac{\pi}{4}\right]=\tan ^{-1}\left[\tan \left(-\frac{\pi}{4}\right)\right]$ where $-\frac{\pi}{4} \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$

$\therefore \tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)=\tan ^{-1}\left[\tan \left(\frac{-\pi}{4}\right)\right]=\frac{-\pi}{4}$