Answer
$-2-\sqrt{3}$
Work Step by Step
Use the subtraction formula for tangent, $\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$.
$\tan\left(-\frac{5\pi}{12}\right)$
$=\tan\left(\frac{\pi}{4}-\frac{2\pi}{3}\right)$
$=\frac{\tan\frac{\pi}{4}-\tan\frac{2\pi}{3}}{1+\tan\frac{\pi}{4}\tan\frac{2\pi}{3}}$
$=\frac{1-(-\sqrt{3})}{1+1*(-\sqrt{3})}$
$=\frac{1+\sqrt{3}}{1-\sqrt{3}}$
$=\frac{1+\sqrt{3}}{1-\sqrt{3}}*\frac{1+\sqrt{3}}{1+\sqrt{3}}$
$=\frac{1+2\sqrt{3}+3}{1-3}$
$=\frac{4+2\sqrt{3}}{-2}$
$=-2-\sqrt{3}$
