Prove that: ^2 8 + ^23 8 + ^25 8 + ^27 8 =2 - Vidyadip

Question

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

✓

Answer

$\text{LHS}=\cos^2\frac{\pi}{8}+\cos^2\frac{3\pi}{8}+\cos^2\frac{5\pi}{8}+\cos^2\frac{7\pi}{8}$
$\cos^2\frac{\pi}{8}+\cos^2\frac{3\pi}{8}+\cos^2\big(\pi-\frac{3\pi}{8}\big)+\cos^2\big(\pi-\frac{\pi}{8}\big)$
$\cos^2\frac{\pi}{8}+\cos^2\frac{3\pi}{8}+\cos^2\frac{3\pi}{8}+\cos^2\frac{\pi}{8}$
$=2\big(\cos^2\frac{\pi}{8}+\cos^2\frac{3\pi}{8}\big)$
$=2\big(\cos^2\frac{\pi}{8}+\cos^2\big(\frac{\pi}{2}-\frac{\pi}{8}\big)\big)$
$=2\big(\cos^2\frac{\pi}{8}+\sin^2\frac{\pi}{8}$
$=2\ \text{RHS}$

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