Question

If cosα + cosβ = 0 = sinα + sin β, then prove that cos2α + cos2β = -2cos (α +β).​

Answer 1

Answer:

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Step-by-step explanation:

sinα+sinβ=a,cosα+cosβ=b

a2+b2=sin2α+sin2β+cos2α+cos2β+2(sinαsinβ+cosαcosβ)

a2+b2=2+2cos(α−β)

∴1+cos(α−β)=2a2+b2

ab=sinαcosα+sinαcosβ+sinβcosα+sinβcosβ

=sin(α+β)+2sin2α+sin2β

=sin(α+β)+sin(α+β)cos(α−β)

=sin(α+β)[1+cos(α−β)]

ab=sin(α+β)(2a2+b2)

∴sin(α+β)=a2+b