Answer
$-90^{o}$
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 $\csc y=-1\quad (\sin y=-1)$
$\displaystyle \sin-\frac{\pi}{2}=\frac{1}{2}$, , that is,
$\csc (-\displaystyle \frac{\pi}{2})=-1$ and $-\displaystyle \frac{\pi}{2}\in [-\displaystyle \frac{\pi}{2},0)\cup(0,\frac{\pi}{2}]$.
So,
$y=-\displaystyle \frac{\pi}{2}$
To convert radians to degrees, multiply y with $\displaystyle \frac{180^{o}}{\pi}$
$\displaystyle \theta=-\frac{\pi}{2}\cdot\frac{180^{o}}{\pi}=-90^{o}$