The expansion about what value of x?
The coefficient of in the Taylor series expansion of f(x) about x= a is where is the nth derivative of f at x= a. Can you find the nth derivative of cos(2x)?
People have. You have refused to do what you have been asked. You say you can get this "to the power of n but not 2n" but your original problem says nothing about a power of 2n. Do you mean you could find the coefficient of x^n if the problem were to expand cos(x) rather than cos(2x)?
(Around, I assume, x= 0. You still haven't answered the very first question asked of you.)
If so, please show us what you would get if this were cos(x) rather than cos(2x).