Complex numbers and trig identities: $\cos(3\theta) + i \sin(3\theta)$ Using the equally rule $a + bi = c + di$ and trigonometric identities how do I make...
$$\cos^3(\theta) - 3\sin^2(\theta)\ \cos(\theta) + 3i\ \sin(\theta)\ \cos^2(\theta) - i\ \sin^3(\theta)=
\cos(3\theta) + i\ \sin(3\theta)$$
Apparently it's easy but I can't see what trig identities to substitute
PLEASE HELP!

From the wording, it appears that you are asked to solve the problem without using DeMoivre's Theorem!
Then you need information about $\cos 3x$ and $\sin 3x$ from another source. Luckily, you probably have all the tools needed. 
For $\cos 3x$, write it as $\cos(2x+x)$, and use the usual cosine of a sum rule. We get $\cos 2x\cos x-\sin 2x\sin x$. Using the double-angle identities, you then get $(\cos^2 x-\sin^2 x)\cos x-2\sin^2 x\cos x$.   I used the rule $\cos 2x=\cos^2 x-\sin^2 x$ because it fits in well with what you are asked to prove. 
We conclude that $\cos 3x=\cos^3 x-3\sin^2 x\cos x$.
You will need similar information about $\sin 3x$. Be guided by the ultimate result you are looking for. But start with $\sin 3x=\sin(2x+x)=\sin 2x \cos x+\cos 2x\sin x$. Of course you will then use $\sin 2x=2\sin x\cos x$, and an appropriate identity for $\cos 2x$.
