Question

How do you find the exact values of $\tan \left(\frac{\mathrm{\pi }}{8}\right)$ using the half-angle formula?

Answer 1

Compute the required value.

Convert the angle from radians to degree.

$$\begin{gathered} \tan \left(\frac{\mathrm{\pi }}{8}\right)=\tan \left(\frac{\mathrm{\pi }\times 180}{8\times \mathrm{\pi }}\right) \\ \Rightarrow \tan \left(\frac{\mathrm{\pi }}{8}\right)=\tan \left(22.5°\right) \end{gathered}$$

We know that, $\tan \alpha =\frac{2\tan {\displaystyle \frac{\alpha }{2}}}{1-{\tan }^2{\displaystyle \frac{\alpha }{2}}}$

Here $\alpha =45°$

$\Rightarrow$ $\tan 45°=\frac{2\tan 22.5°}{1-{\tan }^{2}22.5°}$

Let us assume $\tan \left(22.5°\right)=x$

$\Rightarrow$ $1=\frac{2x}{1-x^2}$ $\left[\because \tan 45°=1\right]$

$\Rightarrow$ $1-x^2=2x$

$\Rightarrow$ $x^2+2x-1=0$

$\Rightarrow$ $x=\frac{-2\pm \sqrt{2^2+4}}{2}$ $\left[\because fora{x}^2+bx+c;x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\right]$

$\Rightarrow$ $x=-1\pm \sqrt{2}$

As $22.5°$ lies in $1^{st}$quadrant and here $\tan$ value is positive, so $-1-\sqrt{2}$ will be ignored.

Therefore, $x=\sqrt{2}-1$

Hence, the value of $\tan \left(22.5°\right)$ is $\sqrt{2}-1$.