I have to make use of:

$\displaystyle sin2x = 2sinxcosx$

$\displaystyle cos2x = cos^2x - sin^2x$

$\displaystyle sin3x = 3sinx - 4sin^3x$

$\displaystyle cos3x = 4cos^3x - 3cosx$

In order to prove:

$\displaystyle sin3x = 4sinxcos^2x - sinx$

$\displaystyle sin4x = 8sinxcos^3x - 4sinxcosx$

ect.

It looks like they are connected but I just can't see how they are connected.

Below is some of my working.

$\displaystyle sin3x = 4sinxcos^2x - sinx$

$\displaystyle RHS = 2(2sinxcosx)cosx - sinx $

$\displaystyle RHS = (sin2x)2cosx - sinx$

I am then promptly stumped

Should I be working with LHS, what sort of substitutions should I be doing?