<s>[INST] Find the solution of the equation Find all real solutions of this equation :
$$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
 [/INST] The function 
$$f(x)=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$ 
is non-negative and decreasing on its domain, $-2\le x\le2$.  Since $f(2)=\sqrt2\lt2$, the equation $x=f(x)$ has exactly one solution.  That solution will be a root of the polynomial of degree $8$ that comes from repeated squaring:
$$((x^2-2)^2-2)^2=2+x$$
L.F.'s comment to Nigel Overmars's answer indicates that $x=2\cos(2\pi/9)$ is a solution, so the upshot here is that it is the only solution.
The solution $x=2\cos(2\pi/9)$ can be verified using standard trig identities:
$$\begin{align}
\sqrt{2+2\cos(2\pi/9)}&=2\cos(\pi/9)\\
\sqrt{2-2\cos(\pi/9)}&=2\sin(\pi/18)\\
&=2\cos(\pi/2-\pi/18)\\
&=2\cos(4\pi/9)\\
\sqrt{2+2\cos(4\pi/9)}&=\cos(2\pi/9)
\end{align}$$
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