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

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

LHS, using identities sinC +sinD = 2sin(C+D)/2 cos(C-D)/2 and

  CosC –cosD = - 2sin(C+D)/2 sin(C-D)/2

(2sin(cos)]sinx + [ -2sin(sin(]cosx

(2sin2x.cosx)sinx + (-2sin2x.sinx)cosx

= 2sin2x.sinx.cosx - 2sin2x.sinx.cosx

= 0

= RHS