Question

Prove that
cos^2(A)+cos^2(A+π/3)+cos^2(A+2π/3) = 3/2​

Answer 1

Answer:

let A=0

let's vertify

=cos^2(0)+cos^2(0+pie/3)+cos^2(0+2pie/3)

=1+(1/4)+(1/4)

=1+(1/2)

=3/2

Answer 2

    LHS

⇒  cos²(A) + cos²(A + π/3) + cos²(A + 2π/3)

⇒  cos²(A) + cos²(A + 60°) + cos²(A + 120°)

⇒  cos²(A) + [cos(A + 60°)]² + [cos(A + 120°)]²

⇒  cos²(A) + [cos(A) · cos(60°) - sin(A) · sin(60°)]² + [cos(A) · cos(120°) - sin(A) · sin(120°)]²

⇒  cos²(A) + [cos(A) / 2 - √3 · sin(A) / 2]² + [- cos(A) / 2 - √3 · sin(A) / 2]²

⇒  cos²(A) + cos²(A) / 4 + 3 sin²(A) / 4 - √3 · sin(A) · cos(A) / 2 + cos²(A) / 4 + 3 sin²(A) / 4 + √3 · sin(A) · cos(A) / 2

⇒  cos²(A) + cos²(A) / 4 + 3 sin²(A) / 4 + cos²(A) / 4 + 3 sin²(A) / 4

⇒  cos²(A) + cos²(A) / 2 + 3 sin²(A) / 2

⇒  cos²(A) + cos²(A) / 2 + (3/2) sin²(A)

⇒  cos²(A) + cos²(A) / 2 + (3/2) (1 - cos²(A))

⇒  cos²(A) + cos²(A) / 2 + 3/2 - 3 cos²(A) / 2

⇒  3/2

⇒  RHS

Hence Proved!