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

Question

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

Answer 1

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.

Answer 2

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