Answer
$\frac{\sqrt{6}-\sqrt{2}}{4}$
Work Step by Step
$cos~\frac{5\pi}{12}$
$=cos~(\frac{3\pi}{12}+\frac{2\pi}{12})$
$=cos~(\frac{\pi}{4}+\frac{\pi}{6})$
$=cos~\frac{\pi}{4}~cos~\frac{\pi}{6} - sin~\frac{\pi}{4}~sin~\frac{\pi}{6}$
$=(\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) - (\frac{\sqrt{2}}{2})(\frac{1}{2})$
$=\frac{\sqrt{6}-\sqrt{2}}{4}$
