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