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

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1 Answer

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

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