${\tan ^{ - 1}}( - 1)$

Let ${\tan ^{ - 1}}( - 1) = x \Rightarrow - 1 = \tan x$
As we know that the range of principal value branch of ${\tan ^{ - 1}}$ is $\left( { - \frac{\pi }{2},\;\frac{\pi }{2}} \right)$

Then, $- 1 = \tan \left( { - \frac{\pi }{4}} \right)where - \frac{\pi }{4} \in \left( { - \frac{\pi }{2},\;\frac{\pi }{2}} \right)$

Hence, the principal value of ${\tan ^{ - 1}}( - 1)\;is\; - \frac{\pi }{4}.$