Simplify sin(x)^2cos(x)^2

Problem

sin2(x)*cos2(x)

Solution

1. Identify the double angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x)

2. Rearrange the identity to solve for the product of sine and cosine: sin(x)*cos(x)=1/2*sin(2*x)

3. Substitute this expression into the original problem by recognizing that sin2(x)*cos2(x)=(sin(x)*cos(x))2

4. Square the substituted expression: (1/2*sin(2*x))2=1/4*sin2(2*x)

5. Apply the power-reduction identity sin2(θ)=(1−cos(2*θ))/2 where θ=2*x to further simplify if needed.

6. Simplify the resulting expression: 1/4⋅(1−cos(4*x))/2=(1−cos(4*x))/8

Final Answer

sin2(x)*cos2(x)=(1−cos(4*x))/8

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