How to prove $F_{n+1}^2 - F_{n-1}^2 = F_{2n} $ by combinatorial-proofs? I'm working on a problem that asks us to prove for any n $\geq$ 1, $F_n$ the Fibonacci number satisfy $F_{n+1}^2 - F_{n-1}^2 = F_{2n}$.
I learned Fibonacci identity $F_{2n} = F_{1}+F_{3}+F_{5}+...+F_{2n-1}$ by combinatorial-proofs but I don't know how to deal with the LHS.
Any help would be appreciated.

Let
$$\mathcal{F} =
\begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix}
$$
and
$$\mathcal{F}_n =
\begin{pmatrix}
F_{n+1}\\
F_n
\end{pmatrix}
$$
we have
$$\mathcal{F}_{n+1} = \mathcal{F}\mathcal{F}_n$$
so
$$\mathcal{F}^n = 
\begin{pmatrix}
F_{n+1} & F_n\\
F_n     & F_{n-1}
\end{pmatrix}
$$
now write $\mathcal{F}^{2n} = \mathcal{F}^n \mathcal{F}^n$
we deduce among other identities
\begin{align}
F_{2n} &= F_nF_{n+1} + F_nF_{n_1}\\
       &= F_n\left(F_{n+1} + F_{n-1}\right)\\
       &= \left(F_{n+1} - F_{n-1}\right)\left(F_{n+1} + F_{n-1}\right)\\
       &= F_{n+1}^2 - F_{n-1}^2
\end{align}
