Answer
$\frac{\sqrt {2+\sqrt 2}}{2}$
Work Step by Step
Use the Half-Angle formula $cos\frac{u}{2}=\pm\sqrt {\frac{1+cos(u)}{2}}$; choosing the positive sign for $\frac{u}{2}=22.5^\circ, u=45^\circ$, we have $cos22.5^\circ= \sqrt {\frac{1+cos(45^\circ)}{2}}=\sqrt {\frac{1+\sqrt 2/2}{2}}=\frac{\sqrt {2+\sqrt 2}}{2}$
