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