Prove by induction that: $F_n\cdot F_{n+2} - F^2_{n+1} = (-1)^{n+1}$

Question

I can't figure out how to solve: $$F_{n} \cdot F_{n+2}- F^2_{n+1}= (-1)^{n+1}$$ I got to the point where I have to assume that: $$F_{k}\cdot F_{k+2} - F^2_{k+1} = (-1)^{k+1}$$ is true and to prove $$F_{k+1} \cdot F_{k+1+2} - F^2_{k+1+1} = (-1)^{k+1+1}$$ is true. I don't know how to approach this problem because of the $$F^2_{k+1+2}$$ term since I don't know if that is multiplying $F_{k+1+2}$ and $F_{k+1+2}$ or if there is another rule that my professor didn't explain to us in class.