If cosα + cosβ =

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)

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