Use the power reducing identities to reduce 7sin^2(x)cos^4(x) to the first power of cosine

Question

 please include all steps, the power reduction formulas are kind of confusing me 

Answer 1

This isn't what you asked for, but I can give you the final answer this way: http://www.wolframalpha.com/input/?i=simplify+7+sin%5E2%28x%29+cos%5E4%28x%29

 

7 sin²(x) cos⁴(x) = (7/32) (2 + cos(2x) - 2cos(4x) - cos(6x))

 

The thing is, I'm not sure what identities you can take as given or not. For example, you can go to https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formula to get

 

sin²(x) cos²(x) = (1 - cos(4x)) / 8

 

This can be proven by the half-angle formulas:

 

sin²(x) = (1 - cos(2x)) / 2

 

cos²(x) = (1 + cos(2x)) / 2

 

To take care of the remaining cos²(x), you'd just have to use the half-angle formula over and over again. It's tedious, but do-able.