Answer
$1$
Work Step by Step
We know that $sec$ has a period of $2\pi$, hence first we try and find a value where the argument is between $\frac{-\pi}{2}$ and $\frac{3\pi}{2}$. Therefore $sec(8\pi)=sec(8\pi-2\pi)=sec(6\pi)=sec(6\pi-2\pi)=sec(4\pi)=sec(4\pi-2\pi)=sec(2\pi)=sec(2\pi-2\pi)=sec(0)=1$
