Answer
$sec(arcsin\frac{5}{13})=\frac{13}{12}$
Work Step by Step
Let $u=arcsin\frac{5}{13}$. Then:
$sin~u=\frac{5}{13}$
The range $arcsin~x$ is $-\frac{\pi}{2}\lt x\lt\frac{\pi}{2}$. So, since $arcsin\frac{5}{13}\gt0$, then $0\lt u\lt\frac{\pi}{2}$ (First Quadrant)
$cos^2u+sin^2u=1$
$cos^2u=1-sin^2u$
$cos^2u=1-(\frac{5}{13})^2=1-\frac{25}{169}=\frac{144}{169}$
$cos~u=\frac{12}{13}~~$ (First Quadrant)
$sec(arcsin\frac{5}{13})=sec~u=\frac{1}{cos~u}=\frac{13}{12}$
