Answer
$-\frac{1}{2}\sqrt{2-\sqrt{2}}$
Work Step by Step
Use the half-angle formula, $\cos\frac{u}{2}=\pm\sqrt{\frac{1+\cos u}{2}}$. Note that $112.5^\circ$ is in Quadrant II, where cosine is negative, so we take the negative square root.
$\cos 112.5^\circ$
$=\cos \frac{225^\circ}{2}$
$=-\sqrt{\frac{1+\cos 225^\circ}{2}}$
$=-\sqrt{\frac{1+(-\frac{\sqrt{2}}{2})}{2}}$
$=-\sqrt{\frac{(1-\frac{\sqrt{2}}{2})*2}{2*2}}$
$=-\sqrt{\frac{2-\sqrt{2}}{4}}$
$=-\frac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}$
$=-\frac{1}{2}\sqrt{2-\sqrt{2}}$
