Answer
$csc(cos^{-1}~\frac{\sqrt 3}{2})=\frac{1}{sin~u}=2$
Work Step by Step
Let $u=cos^{-1}~\frac{\sqrt 3}{2}$. Then:
$cos~u=\frac{\sqrt 3}{2}$
The range $cos^{-1}~x$ is $0\lt x\lt \pi$. So, since $0\lt cos^{-1}~\frac{\sqrt 3}{2}\lt\frac{\pi}{2}$, then $0\lt u\lt\frac{\pi}{2}$ (First Quadrant)
$sin^2u+cos^2u=1$
$sin^2u=1-cos^2u$
$sin^2u=1-(\frac{\sqrt 3}{2})^2=1-\frac{3}{4}=\frac{1}{4}$
$sin~u=\frac{1}{2}$ (First Quadrant)
$csc(cos^{-1}~\frac{\sqrt 3}{2})=csc~u=\frac{1}{sin~u}=2$
