Answer
$\sqrt 2-1$
Work Step by Step
Use the Half-Angle Formula, $tan\frac{\pi}{8}=tan\frac{1}{2}(\frac{\pi}{4})=\sqrt {\frac{1-cos\frac{\pi}{4}}{1+cos\frac{\pi}{4}}}=\sqrt {\frac{1-\frac{\sqrt 2}{2}}{1+\frac{\sqrt 2}{2}}}=\sqrt {\frac{2-\sqrt 2}{2+\sqrt 2}}
=\sqrt {\frac{(2-\sqrt 2)^2}{(2+\sqrt 2)(2-\sqrt 2)}}=\frac{2-\sqrt 2}{\sqrt 2}=\sqrt 2-1$
