Answer
The value of $y$ here is $$y=\frac{\pi}{4}$$
Work Step by Step
$$y=\csc^{-1} \sqrt 2$$
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 cosecant 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 $y=\csc^{-1}\sqrt 2$ into $\csc y=\sqrt 2$
We know that $$\csc\frac{\pi}{4}=\frac{1}{\sin\frac{\pi}{4}}=\frac{1}{\frac{\sqrt 2}{2}}=\frac{2}{\sqrt2}=\sqrt 2$$
And $\frac{\pi}{4}$ 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 $$y=\frac{\pi}{4}$$
