Taylor Series

Use taylor series to verify

i know cos z =

and sin z =

and

but i cant get the steps to get there.

The sum for runs over even terms, and the corresponding one for of odd terms. So you just have to check that the coefficients of for odd/even indexes match.

Start with the series.
Break it down by what happens to that factor.
That will break the series into the sum of four series via the values of n mod 4.
Collect real and imaginary terms.
Simpify for the real and imaginary parts (this takes some care - gotta manipulate the series and their indices and such carefully).
That'll do it.