If cosα + cosβ = 0 = sinα + sinβ, then prove that cos 2α + cos 2β = –2cos (α + β).
[Hint: cosα + cosβ)² – (sinα + sinβ)² = 0]

Question

If cosα + cosβ = 0 = sinα + sinβ, then prove that cos 2α + cos 2β = –2cos (α + β).[Hint: cosα + cosβ)2 – (sinα + sinβ)2 = 0]

Philoid · Accepted Answer

To Prove: cos 2α + cos 2β = –2cos (α + β)

Given,

cosα + cosβ = 0 = sinα + sinβ …(1)

∵ LHS = cos 2α + cos 2β

We know that: cos 2x = cos²x – sin²x

∴ LHS = cos²α – sin²α + (cos²β – sin²β)

⇒ LHS = cos²α + cos²β – (sin²α + sin²β)

∵ a² + b² = (a+b)² – 2ab

⇒ LHS = (cosα + cosβ)² – 2cosα cosβ –(sinα + sinβ)² +2sinα sinβ

⇒ LHS = 0 - 2cosα cosβ -0 + 2sinα sinβ {using equation 1}

⇒ LHS = -2(cosα cosβ – sinα sinβ)

∵ cos (α + β) = cosα cosβ – sinα sinβ

∴ LHS = -2 cos (α + β) = RHS

Hence, cos 2α + cos 2β = –2cos (α + β)