(separating terms with even and odd indices) and , , hence
Equating imaginary parts, we get
where we define the polynomial .
For , the real number satisfies , hence , and , so that the previous equation shows that is a root of .
The degree of is , hence it has at most roots. Furthermore, since , we have , hence , which shows that these roots are distinct. Therefore they are the roots of .
For any polynomial with roots , one can factorize . Expanding this expression (mentally) and gathering the terms of common degree, we see that the term of degree is just and the term of degree is . Thus we have .
In the case of our polynomial, this gives .
On the other hand, we have, for , . Applying this to gives two bounds for , hence for by factorizing a few terms out. Letting go to infinity, both bounds are easily seen to converge toward , hence the conclusion.
Feel free to ask for more clarifications.