Prove of the Cassini-Samson Identity

The Cassini-Samson Identity is an identity that involves the Fibonnaci numbers, it states that $F_{n-1}F_{n+1}-F_{n}^2=(-1)^n$ for $n\in\mathbb{N}$.
We thus prove the theorem by induction:
The base case $n=1$ is easily verified. 
Next, assume $n=k$ and prove for $n=k+1$:
$F_{k}F_{k+2}-F_{k+1}^2$
$=F_{k}(F_{k}+F_{k+1})-F_{k+1}^2$
$=F_{k}^2+F_{k+1}(F_{k}-F_{k+1})$
$=F_{k}^2-F_{k+1}F_{k_1}=-(-1)^k=(-1)^{k+1}$. Which is thus proved.