Answer
$\frac{\sqrt 6-\sqrt 2}{4}$
Work Step by Step
Use Addition Formula with $\frac{5\pi}{12}=\frac{2\pi}{12}+\frac{3\pi}{12}=\frac{\pi}{6}+\frac{\pi}{4}$, we have $cos(\frac{5\pi}{12})=cos(\frac{\pi}{6})cos(\frac{\pi}{4})-sin(\frac{\pi}{6})sin(\frac{\pi}{4})=(\frac{\sqrt 3}{2})(\frac{\sqrt 2}{2})-(\frac{1}{2})(\frac{\sqrt 2}{2})=\frac{\sqrt 6-\sqrt 2}{4}$
