Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles. 3 cos4(x)

Answer 1

Hi Titus Jr I.,

My name is Andrew, I think I may be able to help answer this question.

I am assuming this question is in the form 3 * cos⁴(x) and that you need to reduce this to a form of only cos(x) terms.

This can be done through the use of what is called the half angle formula where:

Half angle formula: cos(x/2) = ± [(1/2)( 1 + cos(x))]^1/2

Alternate form of the Half Angle Formula: cos²(x) = (1/2) (1 + cos(2x))

you can get this form by squaring both sides and multiplying the inside of the cosine by 2. This formula can be applied multiple times as follows:

3 * cos⁴(x) = 3 * cos²(x) * cos²(x)

3 * cos⁴(x) = 3 * (1/2) (1 + cos(2x)) * (1/2) (1 + cos(2x))

factoring out the two 1/2

3 * cos⁴(x) = (3/4) * (1 + cos(2x)) * (1/2) (1 + cos(2x))

multiply the terms out

3 * cos⁴(x) = (3/4) * [ 1 + 2cos(2x) + cos²(2x)]

apply the half angle formula again, here for the 2x you just multiply the "x" term in the cosine by 2

3 * cos⁴(x) = (3/4) * [ 1 + 2cos(2x) + (1/2) (1 + cos(4x))]

multiply the terms out

3 * cos⁴(x) = (3/4) * [ 1 + 2cos(2x) + (1/2) + (1/2)cos(4x))]

3 * cos⁴(x) = (3/4) * [ 3/2 + 2cos(2x) + (1/2)cos(4x))]

you might be done here or you can factor out a 1/2

3 * cos⁴(x) = (3/8) * [ 3 + 4cos(2x) + cos(4x))]

In my education the half angle formula in the alternate form shown here was used more than the one with the square root. I hope I have answered your question! Please comment below if you have any other questions!