I am having a hard time, how do I rewrite 3 cos 4x in terms of cos x?

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Answer 1 · 2018-03-26T03:12:09

In terms of $cosx$ , the expression is equivalent to $24cos^4x-24cos^2x+3$ .

We'll need to use this trigonometric identity, called the "cosine double angle formula":

$cos(2theta)=2cos^2theta-1$

Now here's our expression:

$color(white)=3cos(4x)$

$=3(cos(2*2x))$

$=3(2cos^2(2x)-1)$

$=6cos^2(2x)-3$

$=6(cos(2x))^2-3$

$=6(2cos^2x-1)^2-3$

$=6(2cos^2x-1)(2cos^2x-1)-3$

$=6(4cos^4x-2cos^2x-2cos^2x+1)-3$

$=6(4cos^4x-4cos^2x+1)-3$

$=24cos^4x-24cos^2x+6-3$

$=24cos^4x-24cos^2x+3$

It looks kind of weird, but you can check that they are equivalent by graphing them both and seeing that they have the same graph:

![https://www.desmos.com/calculator](useruploads.socratic.org)

(I had to write $(cos^2x)^2$ instead of $cos^4x$ because Desmos doesn't understand $cos^4x$ .)

That's it. Hope this helped!