Answer
$\frac{1}{2}\sqrt{2+\sqrt{3}}$
Work Step by Step
Use the half-angle formula, $\cos\frac{u}{2}=\pm\sqrt{\frac{1+\cos u}{2}}$. Note that $\frac{\pi}{12}$ is in Quadrant I, where cosine is positive, so we take the positive square root.
$\cos \frac{\pi}{12}$
$=\cos \frac{\frac{\pi}{6}}{2}$
$=\sqrt{\frac{1+\cos \frac{\pi}{6}}{2}}$
$=\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$
$=\sqrt{\frac{(1+\frac{\sqrt{3}}{2})*2}{2*2}}$
$=\sqrt{\frac{2+\sqrt{3}}{4}}$
$=\frac{\sqrt{2+\sqrt{3}}}{\sqrt{4}}$
$=\frac{1}{2}\sqrt{2+\sqrt{3}}$
