Answer
$\frac{\sqrt {u^2-1}}{|u|}$
Work Step by Step
Step 1. Letting $csc^{-1}u=t$, we have $csc(t)=u$ and $-\frac{\pi}{2}\le t \le \frac{\pi}{2}$
Step 2. Form a right triangle with sides $y=1,r=|u|,x=\sqrt {u^2-1}$ and angle $|t|$ facing $y$.
Step 3. As $cos(t)$ is positive in the above interval, we have $cos(csc^{-1}u)=cos(t)=\frac{\sqrt {u^2-1}}{|u|}$