Mark the tick against the correct answer in the following:

Question:

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

 

Solution:

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

$\Rightarrow \tan x=\tan \left(\pi-\frac{\pi}{4}\right)\left(\because \tan \left(\frac{3 \pi}{4}\right)=\tan \left(\pi-\frac{\pi}{4}\right)\right)$

$\Rightarrow \tan x=\tan \left(-\frac{\pi}{4}\right)(\because \tan (\pi-\theta)=\tan (-\theta))$

$\Rightarrow x=-\frac{\pi}{4}$

 

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