Hey FatimaA.
Check your signs for the 2nd last statement: you should have 1/4 - 1/8 - 1/8 cos(4x) if the above line is correct.
Express cos^2 x sin^2 x in terms of cos x raised to the first power.
Here is my first attempt at this. I'd like for someone to look over my steps and help me make sure I did this right.
Using double angle formulas for lowering powers:
(cos^2x)(sin^2x) = ((1+cos2x)/2) ((1-cos2x)/2)
= (1 - cos2x + cos2x - cos^2(2x))/4
= (1 - cos^2(2x))/4
= (1/4) (1 - cos^2(2x))
= (1/4) - (1/4) ((1+cos4x)/2)
= (1/4) - (1/8) + (1/8) cos4x
= (1/8) + (1/8) cos4x as final answer.
(cos^2x)(sin^2x) = ((1+cos2x)/2) ((1-cos2x)/2)
= (1 - cos2x + cos2x - cos^2(2x))/4
= (1 - cos^2(2x))/4
= (1/4) (1 - cos^2(2x))
= (1/4) - (1/4) ((1+cos4x)/2)
= (1/4) - (1/8) - (1/8) cos4x
= (1/8) - (1/8) cos4x