$$\begin{gathered} For 0<\theta <\frac{\pi }{2}, if x=\sum _{n-0}^{\infty }Co{s}^{2n}\theta ,y=\sum _{n-0}^{\infty }{\sin }^{2n}\theta and \\ z=\sum _{n-0}^{\infty }Co{s}^{2n}\theta {\sin }^{2n} \theta then \end{gathered}$$

$\begin{array}{rcl}\mathrm{x}& =& \sum _{\mathrm{n}=0}^{\infty } {\cos }^2 \raisebox{1ex}{$\mathrm{n}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\theta }$}\right.\\ & =& 1+{\cos }^2\mathrm{\theta }+{\cos }^4\mathrm{\theta }+.....\\ & =& \frac{1}{1-{\cos }^2\mathrm{\theta }}=\frac{1}{{\sin }^2\mathrm{\theta }}\\ {\sin }^2\mathrm{\theta }& =& \frac{1}{\mathrm{x}}\\ \mathrm{y}& =& \sum _{\mathrm{n}=0}^{\infty } {\sin }^{2\mathrm{n}}\mathrm{\theta }\\ & =& 1+{\sin }^2\mathrm{\theta }+{\sin }^4\mathrm{\theta }+.....\end{array}$

$\begin{array}{rcl} & =& \frac{1}{1-{\sin }^2\mathrm{\theta }}=\frac{1}{{\cos }^2\mathrm{\theta }}\\ & \Rightarrow & {\cos }^2\mathrm{\theta }=\frac{1}{\mathrm{y}}, \frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}=1 \Rightarrow \mathrm{xy}=1\\ \mathrm{z}& =& \sum _{\mathrm{r}=0}^{\infty } {\sin }^{2\mathrm{r}}\mathrm{\theta } {\cos }^{2\mathrm{r}}\mathrm{\theta }\\ & =& 1+{\sin }^2\mathrm{\theta } {\cos }^2\mathrm{\theta }+{\sin }^4\mathrm{\theta } {\cos }^4\mathrm{\theta }+.....\\ & =& \frac{1}{1-{\sin }^2\mathrm{\theta } {\cos }^2\mathrm{\theta }}\\ \mathrm{r}& =& \frac{1}{1-{\displaystyle \frac{1}{\mathrm{x}}}·{\displaystyle \frac{1}{\mathrm{y}}}}\\ \mathrm{z}& =& \frac{\mathrm{xy}}{\mathrm{xy}-1} \Rightarrow \mathrm{xyz}-\mathrm{z}=\mathrm{xy}\\ \mathrm{xyz}& =& \mathrm{xy}+\mathrm{z}\\ \mathrm{xyz}& =& \mathrm{x}+\mathrm{y}+\mathrm{z}\end{array}$