Deriving an equation using Trig formulas

I'm fairly confident in using the Pythagorean, Sum, Difference, Double Angle and half Angle formulas. The problem is that I struggle with the mental gymnastics to demonstrate sin(3x) = something horrendous, in truth I spend far too much time pursuing blind alleys that look right for far too long.

I can get as far as using the Sum rule to break the Sin(3x) into say

Sin(2x + x) = sin(2x)cos(x)+cos(2x)sin(x) = (2sin(x)cos(x))cos(x)+(cos^2(x)-sin^2(x))sin(x)

But getting it into the final form always seems a step too far.

Can anyone offer some pointers as to how you would go about solving such problems (the only example I have is an assignment question so it would be dishonest of me to post a variation on it for help).