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Question

cos(αβ)+cos(βγ)+cos(γα)=32 then prove that
cosα+cosβ+cosγ=sinα+sinβ+sinγ=0

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Solution

cos(αβ)+cos(βγ)+cos(γα)=32
cosαcosβ+sinαsinβ+cosβcosγ+sinβsinγ+cosγcosα+sinγsinα=32
2cosαcosβ+2sinαsinβ+2cosβcosγ+2sinβsinγ+2cosγcosα+2sinγsinα=3
2cosαcosβ+2sinαsinβ+2cosβcosγ+2sinβsinγ+2cosγcosα+2sinγsinα+sin2αcos2α+sin2β+cos2β+sin2γ+cos2γ=0
(cosα+cosβ+cosγ)2+(sinα+sinβ+sinγ)2=0
cosα+cosβ+cosγ=0
and, sinα+sinβ+sinγ=0.

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