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