Question

cos²x+cos²(x+π/3)+cos²(x-π/3)=3/2

Answer 1

this yours question answer

Answer 2

Question :

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

Answer:

We know that, cos2x = 2cos²x - 1

⇒ 1 + cos2x = 2cos²x

⇒ cos²x = (1+cos2x)/2

Using this :

⇒ (1+cos2x)/2 + [1+cos2(x+π/3)]/2 + [1+cos2(x-π/3)]/2

⇒ ½[3 + cos2x + cos2(x+π/3) + cos2(x-π/3)]

⇒ ½[3 + cos2x + cos(2x+2π/3) + cos(2x-2π/3)]

We know : cos(A+B) + cos(A-B) = 2 cosA cosB

⇒ ½[3 + cos2x + 2 cos2x cos2π/3]

⇒ ½[3 + cos2x + 2 cos2x cos(π-π/3)]

⇒ ½[3 + cos2x + 2 cos2x (-cosπ/3)]

⇒ ½[3 + cos2x - ½×2 cos2x]

⇒ ½[3 + cos2x - cos2x]

⇒ ½ × 3

⇒ 3/2 (RHS)

Hence Proved.