Answer
$\displaystyle \csc^{-1}\frac{\sqrt{2}}{2}$ is not defined.
Work Step by Step
Inverse Cosecant Function:
$y=\csc^{-1}x$ or $y=$ arccsc $x$ means that
$x=\csc y$, for $-\displaystyle \frac{\pi}{2} \leq y \leq \frac{\pi}{2}$ , $y\neq 0$.
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If there exists a y such that $\displaystyle \csc y=\frac{\sqrt{2}}{2}$,
then, sine being reciprocal to csc,
$\displaystyle \sin y=\frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\sqrt{2}}{2}=\sqrt{2}$,
which can not be for any y, since the range of sine is $[-1,1].$
So,
$\displaystyle \csc^{-1}\frac{\sqrt{2}}{2}$ is not defined.
