I want to solve a non linear system of two equations $x$ and $z$. I have two equations : $$ x^2-z^2-2 x z +1 =0,$$ $$ z^2-x^2 -2 x z +1 =0.$$ I know that there are two solutions $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$ and $(\frac{-\sqrt{2}}{2},\frac{-\sqrt{2}}{2})$, but I don't know how to find them.
I tried to substract the first equation with the second one and I get : $$ x^2-z^2+x^2-z^2=0$$ but I think that I do not have to do this operation on nonlinear system.
$$ x^2-z^2-2 x z +1 =0$$ $$ z^2-x^2 -2 x z +1 =0$$
Subtracting equations
$$ x^2-z^2+x^2-z^2=0$$
$$x^2=z^2$$
$$x=\mp z$$
Let $x=z$
Substituting in the first equation $$2x^2=1$$
$$x=z=-\frac{1}{\sqrt2}$$ or $$x=z=\frac{1}{\sqrt2}$$
Let $x=-z$
Substituting in the first equation
$$2x^2=-1$$ no solution