A little triggy

Let $a=\sin \theta$ and $b = \cos \theta$ . Then $f(n) = a^n + b^n$ and $f(2) = a^2+b^2 = \sin^2 \theta + \cos^2 \theta = 1$ . Now consider

$\begin{aligned} (a^n+b^n)(a^2+b^2) & = a^{n+2}+b^{n+2} + a^2b^2(a^{n-2}+b^{n-2}) \\ f(n)\color{#3D99F6}f(2) & = f(n+2)+a^2b^2f(n-2) & \small \color{#3D99F6} \text{Note that }f(2) = 1 \\ \implies f(n)-f(n+2) & = a^2b^2f(n-2) \\ \implies \frac {f(3)-f(5)}{f(5)-f(7)} & = \frac {a^2b^2f(1)}{a^2b^2f(3)} = \boxed {\dfrac {f(1)}{f(3)}} \end{aligned}$

Refer picture given above Please upvote lol