Question

prove that cos²x+cos²(x+π÷3)+cos²(x-π÷3)=3÷2​

Answer 1

we have

LHS

$= \frac{1 + \cos2x }{2} + \frac{1 + \cos(2x + \frac{2\pi}{3} ) }{2} + \frac{1 + \cos(2x - \frac{2x}{3} ) }{2}$

$= \frac{1}{2}[3 + \cos2x + \cos(2x + \frac{2\pi}{3} ) + \cos(2x - \frac{2\pi}{3} ) ]$

$= \frac{1}{2}[3 + \cos2x + 2 \cos2x \cos \frac{2\pi}{3}]$

$= \frac{1}{2} (3 + \cos2x + 2 \cos2x \cos(\pi - \frac{\pi}{3} ))$

$= \frac{1}{2}(3 + \cos2x - 2 \cos2x \cos( \frac{\pi}{3}))$

$= \frac{1}{2} (3 + \cos2x - cosx) = \frac{3}{2}$

RHS