A Rāmāṉujaṉ identity (MathML test)

In a letter to Godfrey Harold Hardy, Srīṉivāsa Rāmāṉujaṉ Aiyaṅkār asserts that:

$$\frac{1}{1+\frac{e^{-2\pi \sqrt{5}}}{1+\frac{e^{-4\pi \sqrt{5}}}{1+\frac{e^{-6\pi \sqrt{5}}}{\ddots }}}}=\left(\frac{\sqrt{5}}{1+\sqrt[5]{5^{3/4}{\left(\frac{\sqrt{5}-1}{2}\right)}^{5/2}-1}}-\frac{\sqrt{5}+1}{2}\right)e^{2\pi /\sqrt{5}}$$