Reciprocal of Hyperbolic Cosine Minus One

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Theorem

$\dfrac 1 {\cosh x - 1} = \dfrac 1 2 \operatorname{csch}^2 \dfrac x 2$


Proof

\(\displaystyle \cosh x\) \(=\) \(\displaystyle 1 + 2 \sinh^2 \frac x 2\) Hyperbolic Cosine Double Angle Formula
\(\displaystyle \iff \ \ \) \(\displaystyle \cosh x - 1\) \(=\) \(\displaystyle 2 \sinh^2 \frac x 2\) subtracting $1$ from both sides
\(\displaystyle \iff \ \ \) \(\displaystyle \frac 1 {\cosh x - 1}\) \(=\) \(\displaystyle \frac 1 2 \frac 1 {\sinh^2 \frac x 2}\) taking the reciprocal of both sides
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \operatorname{csch}^2 \frac x 2\) Definition of Hyperbolic Cosecant

$\blacksquare$