Answer
The value of $y$ in need to find is $$y=-\frac{\pi}{6}$$
Work Step by Step
$$y=\csc^{-1} (-2)$$
First, we see that the domain of inverse cosecant function is $(-\infty,\infty)$. Therefore, in fact when we deal with inverse cosecant function, we do not need to do this checking step.
The range of inverse cotangent function is $[-\frac{\pi}{2},0)\hspace{0.2cm}U\hspace{0.2cm}(0,\frac{\pi}{2}]$. In other words, $y\in[-\frac{\pi}{2},0)\hspace{0.2cm}U\hspace{0.2cm}(0,\frac{\pi}{2}]$.
We can rewrite $y=\csc^{-1}(-2)$ into $\csc y=(-2)$
We know that $$\csc\frac{\pi}{6}=2$$ which means $$-\csc\frac{\pi}{6}=-2$$ $$\cot(-\frac{\pi}{6})=-2$$
And $-\frac{\pi}{6}$ belongs to the range $[-\frac{\pi}{2},0)\hspace{0.2cm}U\hspace{0.2cm}(0,\frac{\pi}{2}]$.
Therefore, the exact value of $y$ here is $$y=-\frac{\pi}{6}$$
