Answer
$cot(arctan\frac{5}{8})=\frac{8}{5}$
Work Step by Step
Let $u=arctan\frac{5}{8}$. Then:
$tan~u=\frac{5}{8}$
The range $arctan~x$ is $-\frac{\pi}{2}\lt x\lt\frac{\pi}{2}$. So, since $arctan\frac{5}{8}\gt0$, then $0\lt u\lt\frac{\pi}{2}$ (First Quadrant)
$cot(arctan\frac{5}{8})=cot~u=\frac{1}{tan~u}=\frac{8}{5}$
