The value of $ an \frac{\pi}{8}$ is

Question

The value of $	an \frac{\pi}{8}$ is

Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Since } 	an 2 	heta=\frac{2 	an 	heta}{1-	an ^2 	heta} \ & 	herefore \quad 	an \frac{\pi}{4}=\frac{2 	an \frac{\pi}{8}}{1-	an ^2 \frac{\pi}{8}} \ & \Rightarrow 1=\frac{2 	an \frac{\pi}{8}}{1-	an ^2 \frac{\pi}{8}} \ & 	ext { Let } y=	an \frac{\pi}{8} \ & \Rightarrow 1=\frac{2 y}{1-y^2} \ & \Rightarrow y^2+2 y-1=0 \ & \Rightarrow y=\frac{-2 \pm \sqrt{4+4}}{2} \ & \Rightarrow y=\frac{-2 \pm 2 \sqrt{2}}{2} \ & \Rightarrow y=-1 \pm \sqrt{2} \ & 	an \frac{\pi}{8}=-1 \pm \sqrt{2} \end{aligned}$ Since $\frac{\pi}{8}$ lies in $1^{	ext {st }}$ quadrant. $\begin{aligned} 	herefore \quad 	an \frac{\pi}{8} & 
eq-1-\sqrt{2} \ 	herefore \quad 	an \frac{\pi}{8} & =-1+\sqrt{2} \ & =\sqrt{2}-1 \end{aligned}$

The value of $\tan \frac{\pi}{8}$ is

Solution