How to prove $$1+\cfrac{1}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-4\pi}}{1+\cfrac{e^{-6\pi}}{\cdots}}}} = \left(\sqrt{5\phi}-\phi\right) e^{2\pi/5}.$$ i dont know how to do it. like if there were repeating terms we put it to be $z$ and then again $1/z$ and solve the quadratic but how to go for this. Note that I am interested if there is a way using elementary knowledge as I am only in grade $12$. Thanks!
2026-03-26 21:25:27.1774560327
Is this well known?
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