1.

and as (*)

2.

Multiply numerator and denominator by:

we have: and then apply (*)

3. Definition of where

( this is not valid in my opinion)

4.

So if it's uniformly convergent (you'd have to go over that more carefully)

we have:

5. assuming it exists

since

Then thus:

You counted applying the Power Series expansion of cos(x), right?