The image you posted gives the same value for sin(22.5°) that skeeter has given. If you are trying to find $\displaystyle \cos(22.5^{\circ})$ then use:
$\displaystyle \cos^2\left(\frac{45^{\circ}}{2} \right)=\frac{1+\cos(45^{\circ})}{2}$
$\displaystyle \cos^2(22.5^{\circ})=\frac{1+\frac{1}{\sqrt{2}}}{2 }=\frac{2+\sqrt{2}}{4}$
$\displaystyle \cos(22.5^{\circ})=\frac{\sqrt{2+\sqrt{2}}}{2}$