Question:
Find the principal value of $\tan ^{-1}(-1)$
Solution:
Let $\tan ^{-1}(-1)=y .$ Then, $\tan y=-1=-\tan \left(\frac{\pi}{4}\right)=\tan \left(-\frac{\pi}{4}\right)$
We know that the range of the principal value branch of $\tan ^{-1}$ is
$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and $\tan \left(-\frac{\pi}{4}\right)=-1$
Therefore, the principal value of $\tan ^{-1}(-1)$ is $-\frac{\pi}{4}$.
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