Answer
The value of $y$ here is $$y=\frac{3\pi}{4}$$
Work Step by Step
$$y=\cot^{-1} (-1)$$
First, we see that the domain of inverse cotangent function is $(-\infty,\infty)$. Therefore, in fact when we deal with inverse cotangent function, we do not need to do this checking step.
The range of inverse cotangent function is $(0,\pi)$. In other words, $y\in(0,\pi)$.
We can rewrite $y=\cot^{-1}(-1)$ into $\cot y=-1$
We know that $$\cot\frac{\pi}{4}=1$$ which means $$-\cot\frac{\pi}{4}=-1$$ $$\cot(-\frac{\pi}{4})=-1$$
(If you need a proof here, take $\cot(-X)=\frac{\cos(-X)}{\sin(-X)}$. We know that $\cos(-X)=\cos X$ and $\sin(-X)=-\sin X$. That means $\cot(-X)=\frac{\cos X}{-\sin X}=-\cot X$).
However, $-\frac{\pi}{4}$ does not belong to the range $(0,\pi)$.
So we must take another similar value but stays in quadrant 1 or 2, which belongs to the range. Since $\cot (X+\pi) =\cot x$, $\frac{3\pi}{4}$ is such a value.
Therefore, the exact value of $y$ here is $$y=\frac{3\pi}{4}$$
