Take a look at this Wikipedia article on trigonometric identities. The sum and difference formulas are in there.

For this problem, start writing out each term as a sum of the previous term and look for a pattern.

We already know the sin(2a) though and it isn't hard to find the cos(2a). Every next term can use the previous term to help expand it. Try writing out a few more terms and see what patterns you can find.