<s>[INST] Using De Moivre's Theorem to prove things So I need to use ($\cos\theta$ + i$\sin\theta$)^4 to prove;
1) $\cos4\theta$ = 8$\cos^4\theta$ - 8$\cos^2\theta$ + 1
2) $\sin4\theta$ = 4$\sin\theta$$\cos\theta$ ($\cos^2\theta$ - $\sin^2\theta$)
The farthest I can get is equating;
$\cos4\theta$ = $\cos^4\theta$ - 6$\cos^2\theta$$\sin^2\theta$ + $\sin^4\theta$
and
$\sin4\theta$ = 4$\cos^3\theta$$\sin\theta$ - 4$\cos\theta$$\sin^3\theta$
How do it prove 1) and 2) from this? (assuming I didn't mess up my expansions and the equating)?
 [/INST] You're almost there. 
For $\cos4\theta$, use that $\sin^2\theta=1-\cos^2\theta$ and simplify.
For $\sin4\theta$, factor out $4\sin\theta \cos\theta$.
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