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Question

If A+B+C=Ļ€ then prove that sinA+sinB+sinC=4cosA2ƗcosB2ƗcosC2


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Solution

Prove the given result

Given that ,

A+B+C=Ļ€

And,(A+B)=Ļ€-C

We have to prove:

sinA+sinB+sinC=4cosA2ƗcosB2ƗcosC2 .

LHS :-

sinA+sinB+sinC=2sinA+B2cosA-B2+sinC[Standardidentity]=2sinĻ€-C2cosA-B2+2sinC2cosC2āˆµsin2x=2sinxcosx=2sinĻ€2-C2cosA-B2+2cosĻ€-A-B2cosC2=2cosC2cosA-B2+2cosA+B2cosC2āˆµsinĻ€2-Īø=cosĪø=2cosC2ƗcosA-B2+cosA+B2Standardidentity=4cosC2ƗcosB2ƗcosA2

āˆ“ LHS=RHS

Hence, proved.


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