Reciprocal of One Plus Hyperbolic Cosine

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Theorem

$\dfrac 1 {1 + \cosh x} = \dfrac 1 2 \operatorname{sech}^2 \dfrac x 2$


Proof

\(\displaystyle \cosh x\) \(=\) \(\displaystyle 2 \cosh^2 \frac x 2 - 1\) 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

$\blacksquare$