Answer
$\sqrt 3-2$
Work Step by Step
Step 1. $tan(-\frac{5\pi}{12})=-tan(\frac{5\pi}{12})=-tan(\frac{3\pi}{12}+\frac{2\pi}{12})=-tan(\frac{\pi}{4}+\frac{\pi}{6})=-\frac{tan(\frac{\pi}{4})+tan(\frac{\pi}{6})}{1-tan(\frac{\pi}{4})tan(\frac{\pi}{6})}=-\frac{1+\frac{\sqrt 3}{3}}{1-\frac{\sqrt 3}{3}}=-\frac{3+\sqrt 3}{3-\sqrt 3}$
Step 2. $cot(-\frac{5\pi}{12})=\frac{1}{tan(-\frac{5\pi}{12})}=-\frac{3-\sqrt 3}{3+\sqrt 3}\times\frac{3-\sqrt 3}{3-\sqrt 3}=-\frac{12-6\sqrt 3}{6}=\sqrt 3-2$
