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Hyperbolic tangent

The identity $\cosh^2x-\sinh^2x=1$ implies
$$
1-\tanh^2x=\operatorname{sech}^2x,
\qquad
\coth^2x-\operatorname{csch}^2x=1.
$$

The identity follows directly from the exponential definitions of the <hyperbolic functions>.

Hyperbolic Pythagorean identity

Catenary from hyperbolic cosine

Hyperbolic function

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Rapidity

Hyperbolic Pythagorean identity

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Hyperbolic Pythagorean identity

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