Extensions of Ramanujan's Cos/Cosh Identity

Question

The Ramanujan Cos/Cosh Identity is stated here as $$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+ \left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}= \frac{2\Gamma^4\left(\frac34\right)}{\pi}$$

Then there is a line:

Equating coefficients of $\theta^0$, $\theta^4$, and $\theta^8$ gives some amazing identities for the hyperbolic secant.

Those identities are given here.

So I have two questions:

1. How do we get those formulas from the Cos/Cosh identity?

2. Are there similar identities? (similar to Cos/cosh identity)