How can I use the power reducing formulas to simplify sin^2 2x cos^2 2x to the first power in terms of cosine?

Question

I understand how it works when it is sin^2 x cos^2 x but I do not know what to do when I am given a 2x.

Answer 1

Hello Angel, a fellow El Pasoan.

 

Using the identities

sin²u = (1-cos(2u))/2

cos²u = (1+cos(2u))/2

 

In this case, we can make it so that u=2x.  This is the value that is doubled in the identity.

sin²(2x)cos²(2x)

= ((1-cos(4x))/2)((1+cos(4x))/2)

= (1-cos²(4x))/4

 

We can take it a step further to get it to the first power.

= (1-(1-cos(8x))/2)/4

= (2/2-(1-cos(8x))/2)/4               Common demoninator

= ((1+cos(8x))/2/4

= (1+cos(8x))/8