Question

Prove that $2\sin \frac{\pi }{8}=\sqrt{2-\sqrt{2}}$.

Answer 1

We have

$\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}$

or, $1-2{\sin }^2\frac{\pi }{8}=\frac{1}{\sqrt{2}}$

or, $1-\frac{1}{\sqrt{2}}$$=2{\sin }^2\frac{\pi }{8}$

or, $2-\sqrt{2}$$=4{\sin }^2\frac{\pi }{8}$

or, $2\sin \frac{\pi }{8}=\sqrt{2-\sqrt{2}}$. [ Since $\sin \frac{\pi }{8}$ is positive]