Find the principal value of tan−1 (−1)

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