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

Answer:

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

Step-by-step 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.