Answer
$\dfrac{\sqrt{{2+\sqrt2}}}{2}$
Work Step by Step
The half-angle Identity for cosine can be expressed as:
$\cos {(\dfrac{\theta}{2})}=\pm\sqrt{\dfrac{1+\cos{\theta}}{2}}$
Therefore,
$\cos{(22.5^{\circ})}=\sqrt{\dfrac{1+\cos{(45^{\circ})}}{2}}\\=\sqrt{\dfrac{1+\dfrac{\sqrt2}{2}}{2}}\\=\sqrt{\dfrac{2+\sqrt2}{4}}\\ =\dfrac{\sqrt{{2+\sqrt2}}}{2}$
