Reciprocal of One Plus Hyperbolic Cosine

$\dfrac 1 {1 + \cosh x} = \dfrac 1 2 \operatorname{sech}^2 \dfrac x 2$
 $\displaystyle \cosh x$ $=$ $\displaystyle 2 \cosh^2 \frac x 2 - 1$ Hyperbolic Cosine Double Angle Formula $\displaystyle \iff \ \$ $\displaystyle 1 + \cosh x$ $=$ $\displaystyle 2 \cosh^2 \frac x 2$ adding $1$ to both sides $\displaystyle \iff \ \$ $\displaystyle \frac 1 {1 + \cosh x}$ $=$ $\displaystyle \frac 1 2 \frac 1 {\cosh^2 \frac x 2}$ taking the reciprocal of both sides $\displaystyle$ $=$ $\displaystyle \frac 1 2 \operatorname{sech}^2 \frac x 2$ Definition of Hyperbolic Secant
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