Mark the tick against the correct answer in the following:
The value of $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$ is
A. $\frac{3 \pi}{4}$
B. $\frac{\pi}{4}$
C. $\frac{-\pi}{4}$
D. none of these
To Find: The value of $\tan ^{-1}\left(\tan \left(\frac{3 \pi}{4}\right)\right)$
Now, let $x=\tan ^{-1}\left(\tan \left(\frac{3 \pi}{4}\right)\right)$
$\Rightarrow \tan x=\tan \left(\frac{3 \pi}{4}\right)$
Here range of principle value of $\tan$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
$\tan ^{-1}\left(\tan \left(\frac{3 \pi}{4}\right)\right)$ is
$\Rightarrow \tan x=\tan \left(\pi-\frac{\pi}{4}\right)\left(\because \tan \left(\frac{3 \pi}{4}\right)=\tan \left(\pi-\frac{\pi}{4}\right)\right)$
$\Rightarrow \tan x=\tan \left(-\frac{\pi}{4}\right)(\because \tan (\pi-\theta)=\tan (-\theta))$
$\Rightarrow x=-\frac{\pi}{4}$