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Question:

Prove that: (sin3x+sinx)sinx+(cos3x-cosx)cosx=0

Solution:

LHS=(sin3x+sinx)sinx+(cos3x-cosx)cosx
=[2sin(3x+x)/2cos(3x-x)/2]sinx+[2sin(3x+x)/2sin(3x-x)/2]cosx
→2[sinxcosxsin2x-sinxcosxsin2x]=0=RHS