Question:
Write the value of $\tan ^{-1}\left\{\tan \left(\frac{15 \pi}{4}\right)\right\}$.
Solution:
We have
$\tan ^{-1}\left\{\tan \left(\frac{15 \pi}{4}\right)\right\}=\tan ^{-1}\left\{\tan \left(4 \pi-\frac{\pi}{4}\right)\right\}$
$\tan ^{-1}\left\{-\tan \left(\frac{\pi}{4}\right)\right\} \quad[\because \tan (4 \pi-x)=-\tan x]$
$=\tan ^{-1}\left\{\tan \left(-\frac{\pi}{4}\right)\right\}$
$=-\frac{\pi}{4} \quad\left[\because \tan ^{-1}(\tan x)=x\right]$
$\therefore \tan ^{-1}\left\{\tan \left(\frac{15 \pi}{4}\right)\right\}=-\frac{\pi}{4}$
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