"Bizarre" continued fraction of Ramanujan! But where's the proof?

Question

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$

"Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued fractions so i have no idea how to attempt to prove it.

Answer 1

The left hand side is $ \tanh \frac{\pi}{2} $ so it may be worth looking at a continued fraction for $ \tanh x $. It looks like a special case of Gauss's continued fraction (I can't link to it directly but there is a Wikipedia page on it). Also, Ramanujan is not mentioned.

Answer 2

(Not an answer but too long for a comment.) Here is another "bizarre" one involving $e$ and $\pi$ (correcting a typo spotted by Paramanand Singh),

$$\sqrt{\frac{\pi\,e^x}{2x}}=1+\frac{x}{1\cdot3}+\frac{x^2}{1\cdot3\cdot5}+\frac{x^3}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{\color{blue}x+\cfrac{1}{\color{blue}1+\cfrac{2}{\color{blue}x+\cfrac{3}{\color{blue}1+\ddots}}}}$$

for $x>0$. No need to mention who found this. As Kevin Brown of Mathpages commented in this old sci.math post, "Is there any other mathematician whose work is instantly recognizable?"