Precalculus, Chapter 5, 5.5, Section 5.5, Problem 28

According to the power reducing formulas, you may re-wrute the expression such that:
sin^4 (2x) = sin^2(2x)*sin^2(2x) = (1 - cos2*(2x))/2*(1 - cos2*(2x))/2
sin^4 (2x) =((1 - cos 4x)^2)/4
sin^4 (2x) =(1 - 2cos 4x + cos^2 4x)/4
sin^4 (2x)= (1 - 2cos 4x + (1 + cos 8x)/2)/4
sin^4 (2x)= (2 - 4cos 4x + 1 + cos 8x)/8
sin^4 (2x)= (3 - 4cos 4x + cos 8x)/8
Hence, eusing the power reducing formulas yields sin^4 (2x)= (3 - 4cos 4x + cos 8x)/8.