Answer
$\dfrac {\sqrt {2}}{4}\left( 1-\sqrt {3}\right) $
Work Step by Step
$\cos \left( \dfrac {7\pi }{12}\right) =\cos \left( \dfrac {\pi }{3}+\dfrac {\pi }{4}\right) =\cos \dfrac {\pi }{3}\cos \dfrac {\pi }{4}-\sin \dfrac {\pi }{3}\sin \dfrac {\pi }{4}=\dfrac {\sqrt {2}}{4}\left( 1-\sqrt {3}\right) $