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