sin(8x) = sin2(4x)

you should know that sin2x = 2sinxcosx. Applying this to the above, you have

sin2(4x) = 2sin(4x).cos(4x)

again,

sin(4x) = sin2(2x) =2 sin(2x)cos(2x), so you have,

sin(8x) = 2 [ 2 sin(2x)cos(2x)] cos(4x) = 4sin(2x)cos(2x)cos(4x)

again put sin2x = 2sinxcosx in the above, and you have your answer