Question

Find the value of :

${\displaystyle \cos \frac{\pi }{8}}$

Answer 1

We know that, cos2θ = 2cos²θ − 1

1 + cos2θ = 2cos²θ

cos²θ = ${\displaystyle \frac{1+\cos 2\theta }{2}}$

Substituting θ = ${\displaystyle \frac{\pi }{8}}$, we get

${\displaystyle {\cos }^2 \frac{\pi }{8}=\frac{1+\cos 2\left(\frac{\pi }{8}\right)}{2}}$

= ${\displaystyle \frac{1+\cos \left(\frac{\pi }{4}\right)}{2}}$

= ${\displaystyle \frac{1+\frac{1}{\sqrt{2}}}{2}}$

= ${\displaystyle \frac{\sqrt{2}+1}{2\sqrt{2}}}$

∴ ${\displaystyle \cos \frac{\pi }{8}=\sqrt{\frac{\sqrt{2}+1}{2\sqrt{2}}} ...[\because \cos \frac{\pi }{8}\text{is posiive}]}$

∴ ${\displaystyle \cos \frac{\pi }{8}=\sqrt{\frac{\sqrt{2}+1}{2\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}}}$

= ${\displaystyle \sqrt{\frac{2+\sqrt{2}}{4}}}$

∴ ${\displaystyle \cos \frac{\pi }{8}=\frac{\sqrt{2+\sqrt{2}}}{2}}$