Answer
$2+\sqrt {3}$
Work Step by Step
$\tan \dfrac {5\pi }{12}=\tan \left( \dfrac {\pi }{4}+\dfrac {\pi }{6}\right) =\dfrac {\tan \dfrac {\pi }{4}+\tan \dfrac {\pi }{6}}{1-\tan \dfrac {\pi }{4}\tan \dfrac {\pi }{6}}=\dfrac {1+\dfrac {\sqrt {3}}{3}}{1-\dfrac {\sqrt {3}}{3}}=\dfrac {3+\sqrt {3}}{3-\sqrt {3}}=$
$=\dfrac {\left( 3+\sqrt {3}\right) \left( 3+\sqrt {3}\right) }{\left( 3-\sqrt {3}\right) \left( 3+\sqrt {3}\right) }=\dfrac {3^{2}+2\times 3\times \sqrt {3}+\sqrt {3}^{3}}{3^{2}-\sqrt {3}^{2}}=\dfrac {12+6\sqrt {3}}{6}=2+\sqrt {3}$
