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7) Identity
sin(2A)+sin(2B)+sin(2C)=4sinAsinBsinC:
sin2A+sin2B+sin2C = 2sin(A+B) cos(A-B)+2sin C cos C
=2sin C(cos(A-B)+cos C), due to the fact that sin(A+B) = sin[180-(A+B)]=sin C
Hence,
=4sin C cos1/2(A+C-B) cos1/2(A-B-C)
=4sin C cos1/2([180-B]-B) cos1/2(A-[180-A])
=4sin C cos(90-B) cos(A-90)
=4sin C cos(90-B) cos(90-A)
=4sinAsinBsinC
Therefore LHS=RHS making it an identity.