Originally Posted by

**Matt Westwood** ... but anyway, back to your original question.

I'd prove the first one by using de Moivre's formula:

$\displaystyle (\cos x + i \sin x)^n = \cos (nx) + i \sin (nx)$

Substitute $\displaystyle n=5$, multiply out the LHS, simplify out all the algebra (replacing every instance of $\displaystyle i^2$ with $\displaystyle -1$, then equating the real and imaginary parts.

If you haven't investigated complex numbers yet, then perhaps now would be a good time to get into them. Come on in, the water's lovely.