Question

Rewrite with only sin x and cos x. cos 3x

Answer 1

cos 3x can be written using the sum and difference identity.
cos (3x) = cos (2x + 1)
cosx * cos2x - sinx * sin2x
next, you have to use the double angle identities for both sine and cosine. there are three options for cosine, so choose one. I'll use
`2cos x^(2)`x - 1.
so, (
`2cos x^(2)`x - 1)cosx - 2sinx(cosxsinx)
you keep using those identities until you come to your final answer of
`4cos^(3)x - 3cosx`

Answer 2

`\cos(3x)=\cos^3(x)-3\sin^2(x)\cos(x)`

Explanation:

`\cos(3x)=\cos(2x+x)`

Then using the identity for the cosine of a sum:

`=\cos(2x)\cos(x) - \sin(2x)\sin(x)`

Then using the identities for the double angle which state that:
`\cos(2x)=\cos^2(x)-\sin^2(x)` and that
`\sin(2x)=2\sin(x)\cos(x)`, our problem becomes:

`=[\cos^2(x)-\sin^2(x)]\cos(x) - [2\sin(x)\cos(x)]\sin(x)`

Then distributing:

`=\cos^3(x)-\sin^2(x)\cos(x)-2\sin^2(x)\cos(x)`

Then combining like terms:

`=\cos^3(x)-3\sin^2(x)\cos(x)`

This is already in terms of cos(x) and sin(x) alone so we can stop.