Question

38. If cosa+cos B+cosy = sina+sinß+siny=0, show that cos3a +cos3 + cosy = 3cos (a+ B +)
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Answer 1

Answer:

cos(β−γ)+cos(α−β)+cos(γ−α)=−

2

3

⇒cosβcosγ+sinβsinγ+cosαcosβ+sinαsinβ+cosγcosα+sinγsinα=−

2

3

⇒(2cosβcosγ+2cosαcosβ+2cosγcosα)+(2sinβsinγ+2sinαsinβ+2sinγsinα)=−3

⇒3+(2cosβcosγ+2cosαcosβ+2cosγcosα)+(2sinβsinγ+2sinαsinβ+2sinγsinα)=−3+3

⇒sin

2

α+cos

2

α+sin

2

β+cos

2

β+sin

2

γ+cos

2

γ+(2cosβcosγ+2cosαcosβ+2cosγcosα)

+(2sinβsinγ+2sinαsinβ+2sinγsinα)=0

⇒(cos

2

α+cos

2

β+cos

2

γ+2cosβcosγ+2cosαcosβ+2cosγcosα)

+(sin

2

α+sin

2

β+sin

2

γ+2sinβsinγ+2sinαsinβ+2sinγsinα)=0

⇒(cosα+cosβ+cosγ)

2

+(sinα+sinβ+sinγ)

2

=0

This is only true when,

cosα+cosβ+cosγ=0

and sinα+sinβ+sinγ=0