You must figure out how to arrange each equation of the system suggesting the use of tangent function.
I propose you to write the top equation such as:
y-y(x^2)=2x (factorization is needed)
y(1-x^2)=2x (divide equation by 1-x^2)
I propose you replacement of x=tan t:
y=2tan t/(1-(tan t)^2)
You are looking at the formula that expresses the tangent function of the angle 2t:y=tan(2t)
I propose you to write the middle equation such as:
z=2y/(1-y^2), where y=tan(2t)
I propose you to write the bottom equation such as:
x=2z/(1-z^2), where y=tan(4t)
use x=tant and x=tan(8t) and equate:
tan t=tan (8t) equivalent to t=8t+npi
use t=-npi/7 to express x,y,z:
Answer: Solution of system: x=tan8npi/7 ; y= tan2npi/7 ; z=tan4npi/7.