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