Answer
$\dfrac{-\pi}{3}$
Work Step by Step
We know that the trigonometric function $\csc^{-1} x$ has range: $[-\dfrac{\pi}{2}, 0) \cup (0, \dfrac{\pi}{2}]$.
Here, $\csc^{-1} (\dfrac{-2 \sqrt 3}{3})=\dfrac{-\pi}{ 3}$, because $\csc(\dfrac{-\pi}{3})=\dfrac{-2 \sqrt 3}{3}$ and $\dfrac{-\pi}{3}$ lies in the range of $\cot^{-1}{x}$ .