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Find the principal value of each of the following :

Question:

Find the principal value of each of the following :

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

Solution:

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

[Formula: $\tan (\pi-x)=-\tan (x)$, as tan is negative in the second quadrant.]

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

[Formula: $\tan ^{-1}(\tan x)=x$ ]

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

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