For positive integer n, let $f\left(n\right)={\sin }^n\theta +{\cos }^n\theta$ . Then $\frac{f\left(3\right)-f\left(5\right)}{f\left(5\right)-f\left(7\right)}=$

$=\frac{{\sin }^3\theta \left(1-{\sin }^2\theta \right)+{\cos }^3\theta \left(1-{\cos }^2\theta \right)}{{\sin }^5\theta \left(1-{\sin }^2\theta \right)+{\cos }^5\theta \left(1-{\cos }^2\theta \right)}$

$=\frac{{\sin }^3\theta {\cos }^2\theta +{\cos }^3\theta {\sin }^2\theta }{{\sin }^5\theta {\cos }^2\theta +{\cos }^5\theta .{\sin }^2\theta }$

$=\frac{{\sin }^2\theta {\cos }^2\theta \left(\sin \theta +\cos \theta \right)}{{\sin }^2\theta {\cos }^2\theta \left({\sin }^3\theta +{\cos }^3\theta \right)}$

$=\frac{\sin \theta +\cos \theta }{{\sin }^3\theta +{\cos }^3\theta }=\frac{f\left(1\right)}{f\left(3\right)}$