please show how solve the system with help of tangent function?2x+(x^2)y=y 2y+(y^2)z=z 2z+(z^2)x=x

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)

y=2x/(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)

z=2tan(2t)/(1-(tan(2t))^2)=tan2*(2t)=tan(4t)

I
propose you to write the bottom equation such
as:

x=2z/(1-z^2), where
y=tan(4t)

x=2tan(4t)/(1-(tan(4t))^2)=tan2*(4t)=tan(8t)

use
x=tant and x=tan(8t) and equate:

tan t=tan (8t) equivalent
to t=8t+npi

t-8t=npi
=>-7t=npi=>t=npi/-7

use t=-npi/7 to express
x,y,z:

Answer: Solution of system: x=tan8npi/7 ; y=
tan2npi/7 ; z=tan4npi/7.