Hello, splash!
It's not clear what definitions and theorems you are allowed.
1) Derive:
I'll assume that we are allowed these two defintions:
. .
Multiply [2] by
We have: .
Add [1]:. .
Then: .
. . Therefore: .
One of the ways of deriving this is extending the definition of the exponent to a complex number.
It can be shown that for a real number, we have,
So we define, an imaginary exponent,
(Withour fear of divergence, this fear is not only convergent but asbsolutely convergent. This fact will enable us to use Riemann's rearrangement theorem).
If we open parantheses we have,
Apply rearrangement and factorize,
You should realize these a power series expanstion for,