Question

Find the principal value of ${\tan }^{-1}(-1).$

Answer 1

Let $y={\tan }^{-1}(-1)$
$\Rightarrow \tan y=-1$
We know that the range of the principal value branch of ${\tan }^{-1}x$ is $(-\frac{\pi }{2},\frac{\pi }{2})$.
$\Rightarrow \tan y=\tan \left(\frac{-\pi }{4}\right)$
$\therefore y=\frac{-\pi }{4}$
Hence, the principal value of ${\tan }^{-1}(-1)$ is $\frac{-\pi }{4}.$