Hi, I could use some help on part (b) of the following question from my Complex Variables, book. I didn't know if this belonged in the Calculus forum or not.

Question 3.2.5

(a) Use the identity

to establish

(b) Use the above identity to also establish

Check the answer by differentiating the series from (a)

Part (a) was easy:

The check in part (b) is easy, too. But I can't do part (b) using the identity given. I tried squaring the identity, but this is as far as I've gotten (essentially, I just eliminated the negative sign):

I've also got this (but I don't know if it's worth anything, or if it is even correct):

I have answered part (a) and the check in part (b) of the following question. But I cannot figure out part (b)