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I am having trouble proving the following identity:

$$\frac{\sinh \tau +\sinh i\sigma }{\cosh \tau +\cosh i\sigma }=-\coth \left(i \frac{\sigma +i\tau }{2}\right)$$

I have tried using identities and the definitions but haven't had much luck. This is a missing step in inverting the bipolar coordinates. Any assistance is appreciated.

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The way I ended up solving it was by applying the same technique as in [ Hint to show $\tanh(z)=\frac{\sinh(2x)+i\sin(2y)}{\cosh(2x)+\cos(2y)}$? ]

To prove the identity: $$\coth z=\frac{\sinh 2x-i \sin 2y}{\cosh 2x-\cos 2y}$$\xo

Then I used the fact that $\sinh (-\tau)=-\sinh \tau $ and $\cosh (-\tau)=\cosh \tau$ to get: $$-\coth \left(\frac{-\tau +i \sigma }{2}\right)=\frac{\sinh \tau+i \sin \sigma}{\cosh \tau-\cos \sigma }$$

Then, use the identities $\sinh z=-i \sin (i z)$ and $\cosh z =\cos (i z)$ to get the identity into the desired form.

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  • $\begingroup$ We can write $\cos\sigma=\cosh(i\sigma)$, so the denominator $\cosh\tau-\cos\sigma$ becomes $\cosh\tau-\cosh(i\sigma)$. The sign does not agree with your target expression. In any case, the identity you're trying to prove is invalid; taking $\sigma=0$, the left-hand side becomes $$\frac{\sinh\tau}{\cosh\tau+1}=\frac{2\sinh(\tau/2)\cosh(\tau/2)}{2\cosh^2(\tau/2)}=\tanh\frac{\tau}{2}$$ but the right-hand side is $$-\coth\left(-\frac{\tau}{2}\right)=\coth\frac{\tau}{2}=\frac{1}{\tanh(\tau/2)}$$ These expressions are almost-never equal. $\endgroup$ – Blue Sep 5 '18 at 22:47
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This appears to be wrong; for example, the case $\sigma=0$ reduces to $\tanh\tau=\coth\frac{\tau}{2}$. One approach is to use $\sinh A+\sinh B =2\sinh\frac{A+B}{2}\cosh\frac{A-B}{2},\,\cosh A+\cosh B=2\cosh\frac{A+B}{2}\cosh\frac{A-B}{2}$ with $A=\tau,\,B=i\sigma$, so the fraction reduces to $\tanh\frac{A+B}{2}=\tanh i\frac{\sigma-i\tau}{2}$. I suspect you may have meant to evaluate$$\frac{\sinh\tau+\sinh i\sigma}{\cosh\tau\mathbf{\color{orange}{-}}\cosh i\sigma}=\frac{2\sinh\frac{A+B}{2}\cosh\frac{A-B}{2}}{2\sinh\frac{A+B}{2}\sinh\frac{A-B}{2}}=\coth\frac{A-B}{2}=-\coth i\frac{\sigma+i\tau}{2}.$$

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