The sech function is a hyperbolic function. It is also known as the hyperbolic secant function.
The sech function is the reciprocal of the cosh function:
$$ \operatorname{sech}{x} = \frac{1}{cosh{x}} $$
(This is analogous with trig functions, where sec is the reciprocal of cos.)
Substituting the formula for cosh gives:
$$ \operatorname{sech}{x} = \frac{2}{e^{x}+e^{-x}} $$
Here is a graph of the function:
We can sketch the sech curve based on the cosh curve:
The cosh curve tends to +∞ for large negative $x$, it has a value of 1 when $x = 0$, and it tends to +∞ for large positive $x$.
The sech curve is the reciprocal of cosh. It tends to zero for large negative $x$, it has a value of 1 when $x = 0$, and it tends to +∞ for large positive $x$.
If we take the two alternative forms of the cosh function, and invert them, we get two alternative forms of the cosech function. First:
$$ \operatorname{cosech}{x} = \frac{2e^{x}}{e^{2x}+1} $$
Alternatively, if we multiply the top and bottom of the original equation for the sinh function by $e^{-x}$ we get:
$$ \operatorname{cosech}{x} = \frac{2e^{-x}}{1+e^{-2x}} $$
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