This identity is not true.

We have:

sin(10x)sin(2x) = (1/2)[cos(10x-2x)-cos(10x+2x)] = (1/2)[cos(8x)-cos(12x)]

sin(8x)sin(4x) = (1/2)[cos(8x-4x)-cos(8x+4x)] = (1/2)[cos(4x)-cos(12x)]

Therefore, sin(10x)sin(2x) is not equal to sin(8x)sin(4x).