Fibonacci identity $F_{n+1}^2 - (F_{n+1}F_n) - F_n^2 = (-1)^n$ I am trying to prove
Let $F_n$ be the $n$th Fibonacci number. Then $F_{n+1}^2 - F_{n+1}F_n - F_n^2 = (-1)^n$
I am not sure where to start with this. 

Observe the truthfulness of the statement for $n=0$: $$1^2 +1\cdot0 -0^2 =1=(-1)^0.$$
We can then show it by induction for all $n$. That is, being true for $k+1$ is implied by the assumption that it's true for $k$:
$$ \begin{align} (F_{k+2})^2 - F_{k+2}F_{k+1} - F_{k+1}^2 &= (F_{k+1} + F_{k})^2 -F_{k+1}(F_{k+1} + F_k)  -(F_{k+1})^2 \\ &= -(F_{k+1})^2 + F_{k+1} F_k + (F_k)^2 \\ &= -(-1)^k = (-1)^{k+1}. \end{align} $$
