^ -1 ( 3 4 ) =? - Vidyadip

Question

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

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Answer

$\tan ^{-1}\left(\tan \frac{3 \pi}{\pi}\right) \neq \frac{3 \pi}{4}$ as $\frac{3 \pi}{4} \notin\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$
$\because \tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$
$=\tan ^{-1}\left[\tan \left(\pi-\frac{\pi}{4}\right)\right]$
$=\tan ^{-1}\left[-\tan \left(\frac{\pi}{4}\right)\right]$
$=-\frac{\pi}{4}$

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