I am looking at examples like this. Yesterday I asked for a proof hoping it would shed some light on its derivation but I am pretty sure the proof given replies on knowing the answer first.
$$2\cos\Big(\frac{2\pi}{7}\Big) = \frac{2+\sqrt[3]{\,7+7\sqrt[3]{\,7+...}}}{1+\sqrt[3]{\,7+7\sqrt[3]{\,7+...}}}$$