The real numbers \(x\), \(y\) and \(z\) satisfy the equations
\[y = \frac{2x}{1-x^2}\,,\qquad z = \frac{2y}{1-y^2}\,,\qquad x = \frac{2z}{1-z^2}\,.\]
Let \(x = \tan\alpha\). Deduce that \(y = \tan 2\alpha\) and show that \(\tan\alpha = \tan 8\alpha\).
Find all solutions of the equations, giving each value of \(x\), \(y\) and \(z\) in the form \(\tan\theta\) where \(-\frac{1}{2}\pi < \theta < \frac{1}{2}\pi\).
Determine the number of real solutions of the simultaneous equations
\[y = \frac{3x - x^3}{1-3x^2}\,,\qquad z = \frac{3y - y^3}{1-3y^2}\,,\qquad x = \frac{3z - z^3}{1-3z^2}\,.\]
Consider the simultaneous equations
\[y = 2x^2 - 1\,,\qquad z = 2y^2 - 1\,,\qquad x = 2z^2 - 1\,.\]
Determine the number of real solutions of these simultaneous equations with \(|x| \leqslant 1\), \(|y| \leqslant 1\), \(|z| \leqslant 1\).
By finding the degree of a single polynomial equation which is satisfied by \(x\), show that all solutions of these simultaneous equations have \(|x| \leqslant 1\), \(|y| \leqslant 1\), \(|z| \leqslant 1\).