Answer
$\sqrt{2}-1$
Work Step by Step
Use the half-angle formula, $\tan\frac{u}{2}=\frac{1-\cos u}{\sin u}$.
$\tan \frac{\pi}{8}$
$=\tan \frac{ \frac{\pi}{4}}{2}$
$=\frac{1-\cos \frac{\pi}{4}}{\sin \frac{\pi}{4}}$
$=\frac{1-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$
$=\frac{(1-\frac{\sqrt{2}}{2})*\sqrt{2}}{\frac{\sqrt{2}}{2}*\sqrt{2}}$
$=\frac{\sqrt{2}-\frac{2}{2}}{\frac{2}{2}}$
$=\frac{\sqrt{2}-1}{1}$
$=\sqrt{2}-1$
