# Reciprocal of Hyperbolic Cosine Minus One

$\dfrac 1 {\cosh x - 1} = \dfrac 1 2 \operatorname{csch}^2 \dfrac x 2$
 $\displaystyle \cosh x$ $=$ $\displaystyle 1 + 2 \sinh^2 \frac x 2$ 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
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