On page 183 of Berndt's Ramanujan's Notebooks Vol. 4, eq. 32.34 reads:
$$ \frac{\sin(2\pi/7)} {\sin^{2}(3\pi/7)} - \frac{\sin(\pi/7)}{\sin^{2}(2\pi/7)} + \frac{\sin(3\pi/7)} {\sin^{2}(\pi/7)} = 2\sqrt{7} $$
If the sines are replaced by hyperbolic sines
$$ \frac{\sinh(2\pi/7)} {\sinh^{2}(3\pi/7)} - \frac{\sinh(\pi/7)} {\sinh^{2}(2\pi/7)} + \frac{\sinh(3\pi/7)} {\sinh^{2}(\pi/7)} \approx 8.196723051395340184\ldots ? $$
Wolfram alpha believes this is a transcendental number.
Is there another analytic expression for it besides the stated one?