Question

Find the principal value of tan-1$(-1)$

Answer 1

Let tan-1$(-1)=y$
Then, $tan\,y=-1=-tan=\bigg(\frac{\pi}{4}\bigg)=tan\bigg(-\frac{\pi}{4}\bigg).$
We know that the range of the principal value branch of
tan-1 is $\bigg(-\frac{\pi}{2},\frac{\pi}{2}\bigg)and\,tan\bigg(-\frac{\pi}{4}\bigg)=-1$
Therefore, the principal value of tan-1$(-1)$ is $-\frac {\pi}{4}$.