Question

use a half-angle identity to find the exact value of tan pi/8

Answer 1

\[\tan x=\frac{\sin x}{\cos x}=\frac{1-\cos2x}{1+\cos2x}~~\implies~~\tan\frac{x}{2}=\frac{1-\cos x}{1+\cos x}\]If \(x=\dfrac{\pi}{4}\), then \(\dfrac{x}{2}=\dfrac{\pi}{8}\), so
\[\tan\frac{\pi}{8}=\frac{1-\cos \dfrac{\pi}{4}}{1+\cos \dfrac{\pi}{4}}=\cdots\]