Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares

Using induction, how can I show the following identity about the fibonacci numbers? I'm having trouble with simplification when doing the induction step.

Identity:

$$f_n^2 + f_{n+1}^2 = f_{2n+1}$$

I get to:

$$f_{n+1}^2 + f_{n+2}^2$$

Should I replace $f_{n+2}$ using the recursion? When I do that, I end up with the product of terms, and that just doesn't seem right. Any guidance on how to get manipulate during the induction step?

Thanks!