Answer
$y=\frac{2\pi}{3}$
Work Step by Step
RECALL:
$y=\sec^{-1}{x} \longrightarrow \cos{y}=\frac{1}{x}$, $y$ is in the interval $[0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]$
Thus, $y=\sec^{-1}{(-2)}$ implies that $\cos{y}=\frac{1}{-2}=-\frac{1}{2}$.
Note that $\cos{(\frac{2\pi}{3})}=-\frac{1}{2}$.
Therefore,
$y=\sec^{-1}{(-2)}\longrightarrow y=\frac{2\pi}{3}$
