^{2}(x) cos

^{2}(x)

^{2}(2x) = 1/4 - (1/4) cos

^{2}(2x) { Pythagorean identity}

^{2}(x) -1 so cos(4x) = 2cos

^{2}(2x) -1

^{2}(2x) = 1/2 + (1/2) cos(4x) . Substitute this into the expression for A

This is for my trig class. Thanks for the help!

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Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...

Let A = sin^{2}(x) cos^{2}(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) sin^{2}(2x) = 1/4 - (1/4) cos^{2}(2x) { Pythagorean identity}

Next the double angle formula for cos is handy cos(2x) = 2cos^{2}(x) -1 so cos(4x) = 2cos^{2}(2x) -1

This can be rearranged to get cos^{2}(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

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