Why not just apply the MacLaurin Series formula to the function cos(x)?
Probably can't because this is in Pre-University Trigonometry.
@kjchauhan
My only problems with this as a "proof" is your original statement: . Was this given?
My other problem is your limit statement: . Was this given as well?
If they are given, it looks great. If they are not given then you have some more work to do.
-Dan