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

$x=1+{\cos }^2\theta +\cos \theta +\mathrm{......}=\frac{1}{1-{\cos }^2\theta }=\frac{1}{si{n}^2\theta }$ 
$y=1+{\sin }^2\theta +\sin \theta +\mathrm{......}=\frac{1}{1-{\sin }^2\theta }=\frac{1}{{\cos }^2\theta }$ 
$z=1+{\sin }^2\theta {\cos }^2\theta +{\sin }^2\theta {\cos }^4\theta +\mathrm{......}=\frac{1}{1-{\sin }^2\theta {\cos }^2\theta }$ 
Now $z=\frac{1}{1-\frac{1}{y}\frac{1}{x}}=\frac{xy}{xy-1}\Rightarrow xyz=x+y+z$
$\because \frac{1}{x}+\frac{1}{y}=1$ 
$x+y=xy$