Find the value of tan frac{pi}{8} | vedclass

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Question

Let $x=\frac{\pi}{8} .$ Then $2 x=\frac{\pi}{4}$

Now $	an 2 x=\frac{2 	an x}{1-	an ^{2} x}$

or $	an \frac{\pi}{4}=\frac{2 	an \frac{\pi}{8}

Vedclass · Accepted Answer

Vedclass · Answer

Let $x=\frac{\pi}{8} .$ Then $2 x=\frac{\pi}{4}$

Now $	an 2 x=\frac{2 	an x}{1-	an ^{2} x}$

or $	an \frac{\pi}{4}=\frac{2 	an \frac{\pi}{8}}{1-	an ^{2} \frac{\pi}{8}}$

Let $y=	an \frac{\pi}{8} .$ 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, $y=	an \frac{\pi}{8}$ is positve. Hence

$	an \frac{\pi}{8}=\sqrt{2}-1$

Find the value of $\tan \frac{\pi}{8}$

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