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 $\frac{3\pi}{8}$ is in Quadrant I, where cosine is positive, so we take the positive square root.
$\cos \frac{3\pi}{8}$
$=\cos \frac{\frac{3\pi}{4}}{2}$
$=\sqrt{\frac{1+\cos \frac{3\pi}{4}}{2}}$
$=\sqrt{\frac{1+(-\frac{\sqrt{2}}{2})}{2}}$
$=\sqrt{\frac{(1-\frac{\sqrt{2}}{2})*2}{2*2}}$
$=\sqrt{\frac{2-\sqrt{2}}{4}}$
$=\frac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}$
$=\frac{1}{2}\sqrt{2-\sqrt{2}}$