Question

Prove that:
$cos ^{2} x + cos ^{2}(x + \frac{\pi}{3} ) + cos ^{2} (x - \frac{\pi}{3} ) = \frac{3}{2}$
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Answer 1

Step-by-step explanation:

$cos^2x+cos^2(x+\frac{\pi}{3})+cos^2(x-\frac{\pi}{3}) \\\to cos^2x+(cos(x+\frac{\pi}{3})+cos(x-\frac{\pi}{3}))^2-2\times cos(x+\frac{\pi}{3})cos(x-\frac{\pi}{3}) \\\to cos^2x+(2cosx\times cos\frac{\pi}{3})^2 - (cos2x+cos\frac{2\pi}{3})\\\to cos^2x+(2cosx\times \frac{1}{2})^2 - cos2x-cos\frac{2\pi}{3}\\\to cos^2x+cos^2x-cos2x-(\frac{-1}{2})\\\to 2cos^2x-cos2x+\frac{1}{2}\\\to 1+\frac{1}{2}\\\to \frac{3}{2}$