Reciprocal of Hyperbolic Cosine Minus One

Theorem

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

Proof

\(\ds \cosh x\)

\(=\)

\(\ds 1 + 2 \sinh^2 \frac x 2\)

Double Angle Formula for Hyperbolic Cosine: Corollary $2$

\(\ds \leadstoandfrom \ \ \)

\(\ds \cosh x - 1\)

\(=\)

\(\ds 2 \sinh^2 \frac x 2\)

subtracting $1$ from both sides

\(\ds \leadstoandfrom \ \ \)

\(\ds \frac 1 {\cosh x - 1}\)

\(=\)

\(\ds \frac 1 2 \frac 1 {\sinh^2 \frac x 2}\)

taking the reciprocal of both sides

\(\ds \)

\(=\)

\(\ds \frac 1 2 \csch^2 \frac x 2\)

Definition 2 of Hyperbolic Cosecant

$\blacksquare$

Also see

• Reciprocal of Hyperbolic Cosine Plus One