Answer
$\frac{1}{u}$
Work Step by Step
Step 1. Letting $sec^{-1}u=t$, we have $sec(t)=u$ and $0\le t \le \pi$
Step 2. Form a right triangle with sides $x=1,r=|u|,y=\sqrt {u^2-1}$.
Step 3. As $cos(t)$ and $sec(t)$ have the same sign in the above interval, we have $cos(sec^{-1}u)=cos(t)=\frac{1}{u}$