Power Reducing Formula

Question

Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles.

sin⁴(x) cos²(x)

Im am truly lost in this question, can anyone help me?

Answer 1

First, you have to get and/or remember those formulas. They are the following:

cos²(u) = 1/2 + cos(2u)/2 and sin²(u) = 1/2 - cos(2u)/2. Then, you can plug these formulas into your original expression to get the following:

sin⁴(x) * cos²(x) = [sin²(u)]² * cos²(u) = (1/2 - cos(2x)/2)² * (1/2 + cos(2x)/2).

Next, you multiply this expression all out, and then when you see a cos²(2x), you use the reduction formula again (but this time with u=2x, so 2u=4x). When you see a cos³(2x), you can do this: cos³(2x) = cos²(2x) * cos(2x), and then use the formula one more time.