<s>[INST] How do you find the real solutions to these simultaneous equations? I am looking for all real $(a,b,c)$ that satisfy the following
\begin{equation}
\left\{
\begin{array}{l}2a + a^2b = b\\ 
2b + b^2c = c\\
2c + c^2a = a\\
\end{array}
\right.
\end{equation}
I know that $a=b=c = 0$ is the only real solution to the problem I know of but I don't know how to prove it. 
I was also given the hint, substitute $a = \tan(x)$. 
 [/INST] $b = \frac {2a}{1-a^2}$
$a = \tan t$
$b = \tan 2t\\
c = \tan 4t\\
a = \tan 8t$
$tan t = tan 8t\\
t + n\pi = 8t\\
7t = n\pi\\
t = \frac {n}{7} \pi$
$a,b,c = \tan \frac {n\pi}{7}, \tan \frac {2n\pi}{7},\tan \frac {4n\pi}{7}$
</s>