John F.

asked • 02/09/17# Express cos^(4)2x in terms of cosine with an exponent of 1?

Hi, this is my first time using this site.

So I have been staring at this trigonometric equation for a long while and looking for what it asks for. For what I know, I think I have to use sum of difference identities, but I am not so sure where to begin from there. I know I am working for an answer of cosine, but then I am lost as to what identity I should use, or if I am using the right identity.

I really appreciate any help for this problem and I thank you.

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## 1 Expert Answer

Jonathan C. answered • 02/09/17

Experienced General Mathematics Tutor

You will need to use the power reducing identity for cosine:

cos

^{2}x = (1+cos(2x))/2 cos

^{2}(2x) = (1+cos(4x))/2 cos

^{4}(2x)= [(1+cos(4x))/2]^{2}Here is our first bout of simplification:

cos

^{4}(2x) = [(1+cos(4x))/2]^{2} cos

^{4}(2x) = (1/4)*[1 + cos(4x)]^{2} cos

^{4}(2x) = (1/4)*(1 + 2cos(4x) + cos^{2}(4x))We are almost there! Use the power reducing identity one more time:

cos

^{4}(2x) = (1/4)*[1 + 2cos(4x) + (1 + cos(2x))/2]Here is our second (and final) bout of simplification:

cos

^{4}(2x) = (1/4)*[1 + 2cos(4x) + (1 + cos(2x))/2] cos

^{4}(2x) = (1/4)*[1 + 2cos(4x) + (1/2)*(1 + cos(2x))] cos

^{4}(2x) = (1/8)*[2 + 4cos(4x) + 1 + cos(2x)] <-- Here I factored out a (1/2) cos

^{4}(2x) = (1/8)*[3 + cos(2x) + 4cos(4x)] <-- Combine like terms and rearrangeThere are several ways that you can rewrite this depending on your teacher's preferences. This one is perfectly valid. I hope this helps.

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Kenneth S.

^{2}x - 1.^{2}x02/09/17