Answer
$\dfrac{2\pi}{3}$
Work Step by Step
$\sec^{-1}{(-2)}=\dfrac{2\pi}{3}$, because $\sec{\left(\frac{2\pi}{3}\right)}=-2$ and $\dfrac{2\pi}{3}$ is in the range of $\sec^{-1}{x}$, which is $\left[0, \dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2}, \pi\right].$