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Use the power reducing identities to write sin^2xcos^2x in terms of the first power of cosine.

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2 Answers

Let   A =  sin2(x) cos2(x)
 
As a fist step we use the fact that sin(x)cos(x) = (1/2) sin(2x)         { double angle formula}
 
Thus  A = (1/4) sin2(2x)  =   1/4 - (1/4) cos2(2x)                            { Pythagorean identity}
 
Next the double angle formula for cos  is handy       cos(2x) = 2cos2(x) -1    so cos(4x) = 2cos2(2x) -1
 
This can be rearranged to get  cos2(2x) = 1/2 + (1/2) cos(4x)    . Substitute this into the expression for A
 
A = 1/4 - 1/8 - (1/8)cos(4x) =   1/8 - (1/8) cos(4x)
 
This expression is in terms of the first power of cosine, so we are done.
Another method:
 
Rearrange and use double angle formula for sine:
sin^2(x)cos^2(x) = ( (1/2)*( 2*sin(x)cos(x) ) )^2
= ( (1/2)*( sin(2x) ) )^2
= (1/4) * sin^2(2x)
 
Use power reducing formula for sine:
= (1/4) *(1/2)*( 1-cos(4x))
= ( 1 - cos(4x) ) / 8