Find the value of $\tan \frac{\pi}{8}$
Find the value of $\tan \frac{\pi}{8}$
Let $x=\frac{\pi}{8} .$ Then $2 x=\frac{\pi}{4}$
Now $\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}$
or $\tan \frac{\pi}{4}=\frac{2 \tan \frac{\pi}{8}}{1-\tan ^{2} \frac{\pi}{8}}$
Let $y=\tan \frac{\pi}{8} .$ Then $1=\frac{2 y}{1-y^{2}}$
or $y^{2}+2 y-1=0$
Therefore $y=\frac{-2 \pm 2 \sqrt{2}}{2}=-1 \pm \sqrt{2}$
since $\frac{\pi}{8}$ lies in the first quadrant, $y=\tan \frac{\pi}{8}$ is positve. Hence
$\tan \frac{\pi}{8}=\sqrt{2}-1$
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