Let $F_k$ be the Fibonacci sequence defined by $F_0=F_1=1$ and $F_k=F_{k-1} + F_{k-2}$
I am trying to show the relation $F_{2k}=(F_{2(k-1)}^2 + 1)/F_{2(k-2)}$ for $k=2,3,4,...$
I first attempted to prove this by induction however I cannot seem to see any simplification. I then used Cassini's identity $F_{n}^2 - F_{n+1}F_{n-1}=(-1)^{n-1}$ to obtain
$(F_{2(k-1)}^2 + 1)/F_{2(k-2)} = (F_{2(k-1)+1}F_{2(k-1)-1} + (-1)^{2(k-1)-1}+1)/F_{2(k-2)}$
$=(F_{2k-1}F_{2k-3}+2)/F_{2k-4}$
However I am similarly stuck with this. Can anyone help with finding a relation which will help me to prove the induction or point me in the right direction.
Your definition of the Fibonacci numbers is not consistent with the customary definition, which is $$\begin{align*} F_0 &= 0, \\ F_1 &= 1, \\ F_n &= F_{n-1} + F_{n-2}. \end{align*}$$ You instead have $F_0 = 1$, which shifts your sequence by $1$. This is the reason for the discrepancy.
We will use the customary definition, in which case the identity to be proven is $$F_{2n+1} = \frac{F_{2n-1}^2 + 1}{F_{2n-3}},$$ which is obtained by substituting $2k = 2n+1$, and is equivalent to the identity $$1 = F_{2n+1} F_{2n-3} - F_{2n-1}^2.$$ A proof by induction is straightforward. Also possible is an approach via the identity $$F_n = \frac{1}{\sqrt{5}} \left( \varphi^n - \bar \varphi^n \right),$$ where $\varphi = (1 + \sqrt{5})/2$ and $\bar \varphi = (1 - \sqrt{5})/2$.