Here is the problem:
Need help please!!
I will give you a push.
I leave the base case to you.
For our inductive step, assume that $\displaystyle F_iF_{i - 2} = (F_{i - 1})^2 + (-1)^i$ (1).
We show that $\displaystyle F_{i + 1}F_{i - 1} = (F_i)^2 + (-1)^{i + 1}$.
Note that
$\displaystyle F_{i + 1}F_{i - 1} = (F_i + F_{i - 1})F_{i - 1}$
$\displaystyle = F_i F_{i - 1} + (F_{i - 1})^2$
Now, change $\displaystyle (F_{i - 1})^2$ using the equation in (1) and simplify to get the desired result.
Try it