If $0<\theta <\frac{\pi }{2}$ and $x=\sum _{n=0}^{\infty }{\cos }^{2n}\theta ,$

$y=\sum _{n=0}^{\infty }{\sin }^{2n}\theta$,

$z=\sum _{n=0}^{\infty }{\cos }^{2n}\theta {\sin }^{2n}\theta$,

then

$$\begin{gathered} x=1+{\cos }^2\theta +{\cos }^4\theta +\mathrm{....}\infty ,\frac{1}{1-{\cos }^2\theta }=\frac{1}{{\sin }^2\theta } similarity \\ y=\frac{1}{{\cos }^2\theta }andz=\frac{1}{1-{\cos }^2\theta {\sin }^2\theta },\frac{1}{{\sin }^2\theta {\cos }^2\theta }+\frac{1}{1-{\sin }^2\theta {\cos }^2\theta } \\ \therefore xyz=\frac{1}{{\sin }^2\theta {\cos }^2\theta (1-{\cos }^2\theta {\sin }^2\theta )} \\ \Rightarrow \frac{1}{{\sin }^2\theta {\cos }^2\theta }+\frac{1}{1-{\sin }^2\theta {\cos }^2\theta }=xy+z \end{gathered}$$