Answer
$-\frac{\sqrt {14}}{2}$.
Work Step by Step
1. Let $sin^{-1}(-\frac{\sqrt 2}{3})=t$, we have $sin(t)=-\frac{\sqrt 2}{3}$ and $t$ in quadrant IV.
2. Let $r=3, y=-\sqrt 2$, we have $x=\sqrt {3^2-(-\sqrt 2)^2}=\sqrt 7$.
3. Thus $cot(sin^{-1}(-\frac{\sqrt 2}{3}))=cot(t)=\frac{x}{y}=-\frac{\sqrt {14}}{2}$.