Answer
$y=\displaystyle \frac{\pi}{4}$
Work Step by Step
$y=\csc^{-1}x$
Domain: $(-\infty, -1]\cup[1, \infty)$
Range: $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$
Quadrants (unit circle): I and IV
---------
$y$ is the number from $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$
such that $\csc y=\sqrt{2}\quad (\displaystyle \sin y=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2})$
$\displaystyle \sin\frac{\pi}{4}=\frac{1}{\sqrt{2}} \Rightarrow \csc (\displaystyle \frac{\pi}{4})=\sqrt{2}$,
and
$\displaystyle \frac{\pi}{4}\in [-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$.
So,
$y=\displaystyle \frac{\pi}{4}$
