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Question

If α+βγ=π, show that sin2α+sin2βsin2γ=2sinαsinβcosγ.

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Solution

L.H.S. =sin2α+sin2βsin2γ
=sin2α+sin(β+γ)sin(βγ)
=sin2α+sin(β+γ)sin(πα)
[α+βγ=π gives βγ=πα]
=sin2α+sin(β+γ)sinα
=sinα[sinα+sin(β+γ)]
sinα[sin(βγ)+sin(β+γ)]
[sinα=sin{π(βγ)}=sin(βγ)]
=sinα2sinβcosγ=2sinαsinβcosγ.

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