The principal value of ${\cot }^{-1}(-1)$ is

The principal value of ${\cot }^{-1}(-1)$ is ${\cot }^{-1}(-1)=y$

Then $\cot y=-1$

$\cot y=-\cot \frac{\pi }{4}=\cot \left[\frac{{\displaystyle -\pi }}{{\displaystyle 4}}\right]=\cot \left[\pi -\frac{{\displaystyle \pi }}{{\displaystyle 4}}\right]=\cot \frac{3\pi }{4}$

The principal value of ${\cot }^{-1}$ in $(0,\pi )=\frac{3\pi }{4}$