Prove that: cos²x + cos²(x + π/3) + cos²(x − π/3) = 3/2

Question

Shiksha Nation · Accepted Answer

L.H.S. = (1 + cos(2x))/2 + (1 + cos(2x + 2π/3))/2 + (1 + cos(2x − 2π/3))/2

= (1/2)[3 + cos(2x) + cos(2x + 2π/3) + cos(2x − 2π/3)]

= (1/2)[3 + cos(2x) + 2·cos(2x)·cos(2π/3)]

= (1/2)[3 + cos(2x) + 2·cos(2x)·cos(π − π/3)]

= (1/2)[3 + cos(2x) − 2·cos(2x)·cos(π/3)]

= (1/2)[3 + cos(2x) − cos(2x)] = 3/2 = R.H.S.

Verified Answer