Answer
$3$
Work Step by Step
Step 1. Letting $csc^{-1}(\sqrt {10})=t, 0\lt t\lt \frac{\pi}{2}$, we have $csc(t)=\sqrt {10}$ and $sin(t)=\frac{\sqrt {10}}{10}$.
Step 2. Form a right triangle with sides $y=\sqrt {10}, r=10, x=3\sqrt {10}$ with $|t|$ facing $y$. We have $cot(t)=3$
Step 3. Thus $cot(csc^{-1}(\sqrt {10}))=cot(t)=3$
