sin(A+B)≡sin(A)cos(B)+cos(A)sin(B)

sin(A-B)≡sin(A)cos(B)-cos(A)sin(B)

sin(A+B)+sin(A-B)≡2sin(A)cos(B)

Let A+B=4x and A-B=2x, then A=3x and B=x, so sin(4x)+sin(2x)≡2sin(3x)cos(x).

sin(6x)≡2sin(3x)cos(3x).

sin(2x)+sin(4x)-sin(6x)≡

2sin(3x)cos(x)-2sin(3x)cos(3x)=

2sin(3x)(cos(x)-cos(3x)).

cos(A+B)≡cos(A)cos(B)-sin(A)sin(B)

cos(A-B)≡cos(A)cos(B)+sin(A)sin(B)

cos(A-B)-cos(A+B)≡2sin(A)sin(B).

Let A-B=x and A+B=3x, then A=2x and B=x, so cos(x)-cos(3x)≡2sin(2x)sin(x).

Therefore:

2sin(3x)(cos(x)-cos(3x))≡

sin(2x)+sin(4x)-sin(6x)≡

(2sin(3x))(2sin(2x)sin(x))=

4sin(x)sin(2x)sin(3x) QED

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