find solutions

\text{i think there is infinit solution of(x,y,z) satifying it }\\ \ \ \text{ put z =}\tan\alpha\\ \text{u will get x=}2\sin^2\alpha\\ \text{similarly take }(x,y)=(\tan\beta,2\sin^2\beta)\\ \text{and} (y,z) = (\tan\gamma ,2\sin^2\gamma) \\ \text{solving these trigonometic equaltion we get } \\ \frac{64\sin^8\beta}{\left(1+4sin^4\beta \right)^2}=\frac{\tan\beta}{2-\tan\beta} \\ \text{which seems to have solution}