Prove that!!!!!!!!!!!!!!!!!!!!!!

Question

sin(3x) + sin(2x) - sin(x)=4sinx.cos(x/2).cos(3x/2)

Answer 1

sin (3x) + sin ( 2x) - sin (x)  = [ sin(3x) + sin (2x) ]-sinx    

      Apply sinα + sinß = 2 sin(α+ß)/2. cos(α-ß)/2

     [ 2sin(3x+2x)/2 .cos(3x-2x)/2 ] - sinx

  [ 2sin 5x/2  cosx/2 ] - sinx      [ Multiply numerator and denominator by 2 ]

 [ 2sin5x/2 cosx/2 ] - sin 2(x)/2    [  sin 2x = 2sinx cosx ]

2sin5x/2 cosx/2 - 2sinx/2 cosx/2

2cosx/2 [ sin5x/2 -sinx/2 ]                [ sinα-sinß =2sin(α-ß/2) cos(α+ß/2) ]

2cosx/2 [2 sin(5x/2-x/2)/2 cos(5x/2+x/2)/2 ]

2cosx/2 [ 2sin(4x/2)/2 cos(6x/2)/2 ]

2cosx/2 [ 2sinx cos3x/2 ]

cosx/2 4sinx cos3x/2

Therefore sin(3x) +sin(2x) -sin(x) = 4sinx.cos(x/2).cos(3x/2).