Request for a proof of the following continued-fraction identity

arithmetic pi recreational-mathematics continued-fractions problem-solving

I have been poring over many texts about continued fractions, but none of them seem to be helping me to prove the following beautiful continued-fraction identity (I am nowhere close): $$ \cfrac{1}{\dfrac{5}{2 \pi} - \cfrac{1}{\dfrac{15}{2 \pi} - \cfrac{1}{\dfrac{25}{2 \pi} - \cfrac{1}{\dfrac{35}{2 \pi} - \cfrac{1}{\dfrac{45}{2 \pi} - \ddots}}}}} = \sqrt{5 + 2 \sqrt{5}}. $$ I warmly welcome anyone in the community to offer any insights that he/she might have. Thank you very much!


Consider the more general continued fraction

$$\cfrac{1}{x - \cfrac{1}{3x- \cfrac{1}{5x- \cfrac{1}{7x - \cfrac{1}{9x - \ddots}}}}}$$

Some numerical experimentation suggests that this is equal to $\tan (1/x)$. Then we can substitute $x = \frac{5}{2 \pi}$ to get the original identity.

Edit: This identity is the third continued fraction identity for tangent listed on this page.

Edit 2: A proof of a continued fraction for $\tan x$, taken from Chrystal's Algebra. To derive the identity above, substitute $1/x$ for $x$ in the identity on that page and then do some manipulations.

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