Solution:

Option(D) is correct.
To Find: The Principle value of $\cot ^{-1}(-1)$
Let the principle value be given by $x$
Now, let $x=\cot ^{-1}(-1)$
$\Rightarrow \cot x=-1$
$\Rightarrow \cot x=-\cot \left(\frac{\pi}{4}\right)\left(\because \cot \left(\frac{\pi}{4}\right)=1\right)$
$\Rightarrow \cot x=\cot \left(\pi-\frac{\pi}{4}\right)(\because-\cot (\theta)=\cot (\pi-\theta))$
$\Rightarrow x=\frac{3 \pi}{4}$