#cot pi/2 - x formula

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Solve cot(pi/2-theta) | cot(pi/2 -x) | cot pi/2 - x formula, Find value cot pi by 2 - x 

Hi friends.. In this tutorial Find the Value of cot(pi/2-theta) cot(pi/2 -x)cot pi/2 - x formula Find the value of cot pi by 2 -xTRIGONOMETRIC IDENTITIESExp...

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A formula of Ramanujan for $\cot\sqrt {w\alpha} \coth\sqrt{w\beta} $

While trying to answer this question I stumbled on a paper by Bruce C. Berndt which contains the following formula by Ramanujan $$\frac{\pi}{2}\cot\sqrt{w\alpha}\coth\sqrt{w\beta}=\frac{1}{2w}+\sum_{m=1}^{\infty} \left(\frac{m\alpha\coth m\alpha} {w+m^2\alpha} +\frac{m\beta\coth m\beta} {w-m^2\beta} \right) \tag{1}$$ which is supposed to hold for all positive numbers $\alpha, \beta$ with $\alpha\beta=\pi^2$.

Berndt mentions that this formula is wrong and missing a term. The corrected version stands as $$\frac{\pi}{2}\cot\sqrt{w\alpha}\coth\sqrt{w\beta}=\frac{1}{2w}+\frac{1}{2}\log\frac{\beta}{\alpha}+\sum_{m=1}^{\infty} \left(\frac{m\alpha\coth m\alpha} {w+m^2\alpha} +\frac{m\beta\coth m\beta} {w-m^2\beta} \right) \tag{2}$$ for $\alpha>0<\beta,\alpha\beta=\pi^2$. Bruce gives some references which contain a proof of the above formula or its equivalents.

Bruce himself derives the above identity by using a change of variables in the following identity established by R. Sitaramchandrarao $$\pi^2xy\cot (\pi x) \coth (\pi y) =1+\frac{\pi^2}{3}(y^2-x^2)-2\pi xy\sum_{n=1}^{\infty} \left(\frac{y^2\coth (\pi n x/y)} {n(n^2+y^2)}-\frac{x^2\coth(\pi n y/x)}{n(n^2-x^2)}\right) \tag{3}$$ Ramanujan gave a similar (but wrong) formula and Sitaramachandrarao fixed it to arrive at $(3)$.

The derivation of $(2)$ from $(3)$ is not that difficult. The RHS of $(3)$ is modified using the identities $$\frac{y^2} {n(n^2+y^2)}=\frac{1}{n}-\frac{n}{n^2+y^2},\frac{x^2}{n(n^2-x^2)}=\frac{n}{n^2-x^2}-\frac{1}{n}$$ and $$\coth z =1+\frac{2}{e^{2z}-1}$$ The derivation also involves a transformation formula for logarithm of Dedekind eta function. However the proof of $(3)$ is omitted in Berndt's paper.

Unfortunately I have not been able to find those references online which contain a proof for $(2)$ or $(3)$. It is also mentioned that the formula could be proved using Mittag-Leffler expansion but I am barely a novice in complex analysis.

It is desirable to find a direct proof of the above result $(2)$ (or $(3)$) which avoids complex analytic methods. I tried to multiply the partial fractions of $\cot a$ and $\coth b$ but I could not manage to get the desired result.

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Solve sin(pi/2+theta) | sin(pi/2+ x) | sin pi/2 + x formula, Find value sin pi by 2 + x

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