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

$x=1+{\cos }^2\theta +{\cos }^4\theta +\mathrm{.....}=\frac{1}{1-{\cos }^2\theta }=\frac{1}{{\sin }^2\theta }$

$y=1+{\sin }^2\varphi +{\sin }^4\varphi +\mathrm{......}=\frac{1}{1-{\sin }^2\varphi }=\frac{1}{{\cos }^2\varphi }$

$Z=1+{\cos }^2\theta {\sin }^2\varphi +{\left({\cos }^2\theta {\sin }^2\varphi \right)}^2+\mathrm{.....}=\frac{1}{1-{\cos }^2\theta {\sin }^2\varphi }$

$Z=\frac{1}{1-(1-1/x)(1-1/y)}=\frac{xy}{x+y-1}$

$\Rightarrow z(x+y)-z=xy\Rightarrow z(x+y)=xy+z$