$\tan \frac{\pi}{8}$ का मान ज्ञात कीजिए।

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

$\tan \frac{\pi}{8}=\sqrt{2}-1$