\[\cos{15{^\circ}} = \sqrt{\frac{1 + \cos{30{^\circ}}}{2}} =\]
\[= \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} =\]
\[= \frac{\sqrt{2 + \sqrt{3}}}{2}\]