Answer
$\frac{\sqrt{2+\sqrt2}}{2}.$
Work Step by Step
By the identity $\cos{(\frac{\alpha}{2})}=\sqrt{\frac{1+\cos{(\alpha)}}{2}}$.
Hence $\cos{(22.5^{o})}=\sqrt{\frac{1+\cos{(45^{o})}}{2}}=\sqrt{\dfrac{1+\frac{\sqrt2}{2}}{2}}=\sqrt{\frac{2+\sqrt2}{4}}=\frac{\sqrt{2+\sqrt2}}{2}.$
