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

$\begin{array}{l}x=\frac{1}{{\cos }^2\theta }=\frac{1}{1-{\sin }^2\theta }\Rightarrow {\sin }^2\theta =\frac{1}{x}\\ y=\frac{1}{1-{\sin }^2\theta }\Rightarrow \frac{1}{{\cos }^2\theta }\Rightarrow {\cos }^2\theta =\frac{1}{y}\\ {\sin }^2\theta +{\cos }^2\theta =1\\ \Rightarrow \frac{1}{x}+\frac{1}{y}=1\Rightarrow x+y=xy\to \left(1\right)\\ z=\frac{1}{1-{\cos }^2\theta {\sin }^2\theta }=\frac{1}{1-\frac{1}{xy}}=\frac{xy}{xy-1}\end{array}$
$\begin{array}{l}\Rightarrow xyz-z=xy\\ \Rightarrow xyz=xy+z\left(\text{ from }\right(1\left)\right)\\ \Rightarrow (x+y)z=xy+z\\ \Rightarrow xz+yz=xy+z\end{array}$