Hyperbolic Secant — Definition, Formula & Examples

The hyperbolic secant, written sech(x), is the reciprocal of the hyperbolic cosine. It equals

\frac{2}{e^x + e^{-x}}

and produces a bell-shaped curve that peaks at 1 when

x = 0

For all real numbers $x$ , the hyperbolic secant is defined as $\operatorname{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}$ . Its range is $(0, 1]$ , it is an even function, and it is continuous and differentiable on $\mathbb{R}$ .

Key Formula

\operatorname{sech}(x) = \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}}

Where:

$x$ = Any real number
$e$ = Euler's number, approximately 2.71828

Worked Example

Problem: Evaluate sech(0) and sech(ln 2).

sech(0): Substitute x = 0 into the formula.

\operatorname{sech}(0) = \frac{2}{e^0 + e^{0}} = \frac{2}{1 + 1} = 1

sech(ln 2): Substitute x = ln 2. Note that

e^{\ln 2} = 2

and

e^{-\ln 2} = \frac{1}{2}

\operatorname{sech}(\ln 2) = \frac{2}{2 + \frac{1}{2}} = \frac{2}{\frac{5}{2}} = \frac{4}{5} = 0.8

Answer: sech(0) = 1 and sech(ln 2) = 4/5.

Why It Matters

The sech function describes the shape of a hanging chain (catenary) cross-section and appears in the soliton solutions of the Korteweg–de Vries equation in physics. Its derivative,

-\operatorname{sech}(x)\tanh(x)

, shows up frequently in calculus exercises on hyperbolic differentiation. In machine learning, sech-based activation functions are sometimes studied as smooth alternatives to other nonlinearities.

Common Mistakes

Mistake: Confusing sech(x) with 1/sinh(x).

Correction: The reciprocal of sinh(x) is csch(x) (hyperbolic cosecant). Sech(x) is specifically 1/cosh(x).