#### Answer

$-30^{o}$

#### Work Step by Step

Solve for radians, then convert to degrees.
$y=\csc^{-1}x$
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 $\csc y=-2\quad (\displaystyle \sin y=-\frac{1}{2})$
In quadrant I, $\displaystyle \sin\frac{\pi}{6}=\frac{1}{2}$,
In quadrant IV, $\displaystyle \sin(-\frac{\pi}{6})=-\frac{1}{2}$, 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,
$y=-\displaystyle \frac{\pi}{6}$
To convert radians to degrees, multiply y with $\displaystyle \frac{180^{o}}{\pi}$
$\displaystyle \theta=-\frac{\pi}{6}\cdot\frac{180^{o}}{\pi}=-30^{o}$