Answer
$2+\sqrt{3}$
Work Step by Step
Use the half-angle formula, $\tan\frac{u}{2}=\frac{1-\cos u}{\sin u}$.
$\tan \frac{5\pi}{12}$
$=\tan \frac{ \frac{5\pi}{6}}{2}$
$=\frac{1-\cos \frac{5\pi}{6}}{\sin \frac{5\pi}{6}}$
$=\frac{1-(-\frac{\sqrt{3}}{2})}{\frac{1}{2}}$
$=\frac{(1+\frac{\sqrt{3}}{2})*2}{\frac{1}{2}*2}$
$=\frac{2+\sqrt{3}}{1}$
$=2+\sqrt{3}$
