Question

What is the equivalent expression for cos^3(2x) that does not involve any powers of cosine or sine greater than 1?

Answer 1

The expression is equal to `(3cos(2x)+cos(6x))/4`.

Explanation:

Use the cosine triple angle identity, and move around the terms to isolate `cos^3x` to get a new identity:

`cos3theta=4cos^3theta-3costheta`

`cos3theta-4cos^3theta=-3costheta`

`-4cos^3theta=-3costheta-cos3theta`

`4cos^3theta=3costheta+cos3theta`

`cos^3theta=(3costheta+cos3theta)/4`

Now use this formula with our expression:

`cos^3(2x)=(3cos(2x)+cos(3*2x))/4`

`color(white)(cos^3(2x))=(3cos(2x)+cos(6x))/4`

You can verify this by graphing `cos^3(2x)` and `(3cos(2x)+cos(6x))/4` and seeing that they have the same curve:

Hope this helped!