Answer
$y=-\displaystyle \frac{\pi}{6}$
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
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$y$ is the number from $[-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$
such that $\csc y=-2\quad (\displaystyle \sin y=-\frac{1}{2})$
Since $\displaystyle \sin\frac{\pi}{6}=\frac{1}{2}$,
$\displaystyle \sin(-\frac{\pi}{6})=-\frac{1}{2}$, that is, $\csc (-\displaystyle \frac{\pi}{6})=-2$,
and,
$-\displaystyle \frac{\pi}{6}\in [-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$,
so,
$y=-\displaystyle \frac{\pi}{6}$
