Question

Prove the following identities:
${\cos }^2\frac{\pi }{8}+{\cos }^2\frac{3\pi }{8}+{\cos }^2\frac{5\pi }{8}+{\cos }^2\frac{7\pi }{8}=2$

Answer 1

L.H.S.

${\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{5\pi }{8}+{\sin }^2\frac{7\pi }{8}$

$={\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2(\pi -\frac{3\pi }{8})+{\sin }^2(\pi -\frac{\pi }{8})$

$={\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\cos }^2\frac{3\pi }{8}+{\cos }^2\frac{\pi }{8}$

$=({\sin }^2\frac{\pi }{8}+{\cos }^2\frac{\pi }{8})+({\sin }^2\frac{3\pi }{8}+{\cos }^2\frac{3\pi }{8})$

$=1+1=2$

R.H.S.