These are pretty standard but if the intent is to "discover" these on your own, just directly apply the formula for the Maclaurin series (a taylor series centred around a = 0):
Taking a = 0:
For the f(x) = cos x, you would need to find:
, , ..
Then generalize a pattern and that should be your Maclaurin series (Of course, if you wanted to prove it you can do it with induction to verify your series).
Also, a google search for maclaurin series would probably give you these results.