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Question

Prove that sin2x+2sin4x+sin6x=4cos2xsin4x

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Solution

L.H.S.=sin2x+2sin4x+sin6x
=(sin6x+sin2x)+2sin4x
sinA+sinB=2sin(A+B2)cos(AB2)
Let A=2x,B=6x
L.H.S=2sin(2x+6x2)cos(2x6x2)+2sin4x
=2sin(4x)cos(2x)+2sin4x
=2sin(4x)(1+cos(2x))=2sin(4x)(2cos2x)
=4cos2xsin4x=R.H.S
Hence Proved.

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