$x=\sum _{n=0}^{\infty }{\cos }^{2n}\theta$

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

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

Let $x=\frac{1}{1-{\cos }^2\theta }=\frac{1}{{\sin }^2\theta }$

$\Rightarrow {\sin }^2\theta =\frac{1}{x} ...\left(1\right)$

$y=\frac{1}{1-{\sin }^2\theta }=\frac{1}{{\cos }^2\theta }\Rightarrow {\cos }^2\theta =\frac{1}{y} ...\left(2\right)$

Adding equations $\left(1\right) \& \left(2\right)$,

$\Rightarrow \frac{1}{x}+\frac{1}{y}=1$

$\Rightarrow x+y-xy=0$

and $z=\frac{1}{1-{\cos }^2\theta {\sin }^2\theta }=\frac{1}{1-{\displaystyle \frac{1}{xy}}}=\frac{xy}{xy-1}$

$\Rightarrow xy z-z=xy$

$\Rightarrow \left(x+y\right)z=xy+z$

$\therefore xz+yz=xy+z$