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Question

If A+B+C=π, then prove that sin2A+sin2B+sin2C=4sinAsinBsinC

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Solution

Consider the problem

A+B+C=π

sin2A+sin2B+sin2C

=2sin(A+B)cos(AB)+2sinCcosC=2sin(πC)cos(AB)+2sinCcosC=2sinCcos(AB)+2sinCcosC=2sinC(cos(AB)+cos(π(A+B)))=2sinC(cos(AB)cos(A+B))=2sinC(2sinBsinA)=4sinAsinBsinC

Hence, LHS=RHS

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