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I'm interested in Ramanujan's identity $$\left ( 1+2\sum_{n=1}^\infty \frac{\cos(n\theta)}{\cosh(n\pi)} \right )^{-2} + \left (1+2\sum_{n=1}^\infty \frac{\cosh(n\theta)}{\cosh(n\pi)} \right )^{-2} = \frac {2 \Gamma^4 \left ( \frac34 \right )}{\pi} = \frac{8\pi^3}{\Gamma^4 \left ( \frac14 \right )},$$ which is shown in here. But how can we proof it in a easy way?

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