co.combinatorics - fibonacci identity using generating function

There are many nice ways of showing that $f_0^2+f_1^2+cdots+f_n^2=f_{n+1}f_n$. I was wondering if there is a way of showing this using the generating function $F(x)=frac{1}{1-x-x^2}=sum_{igeq0}f_ix^i$. In other words, is there any operation (perhaps the Hadamard product) that can be applied to $F(x)$ that would yield the identity above?

What about other identities that involve sums and squares, like $f_1f_2+cdots +f_nf_{n+1}=f_{n+1}^2$ for $n$ odd?

• Next fields - transcendental Galois theory
• Previous books - Errata for Emil Artin's 'The Gamma Function'?