Factsheet: Hyperbolic identities
These are common definitions and identities for hyperbolic functions. For derivatives and antiderivatives, please see Factsheet: List of derivatives and Factsheet: List of integrals respectively.
Definitions of hyperbolic functions
For all real numbers \(x\):
\[\begin{aligned} \cosh(x) &= \frac{e^x + e^{-x}}{2} \\[0.5em] \sinh(x) &= \frac{e^x - e^{-x}}{2} \\[0.5em] \tanh(x) &= \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \\[0.5em] \textrm{coth}(x) &= \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}} \\[0.5em] \textrm{sech}(x) &= \frac{1}{\cosh(x)} = \frac{2}{e^x + e^{-x}} \\[0.5em] \textrm{csch}(x) &= \frac{1}{\sinh(x)} = \frac{2}{e^x - e^{-x}} \end{aligned}\]
Hyperbolic identities
Pythagorean formulas
For all real numbers \(x\):
\[\begin{aligned} \cosh^2(x) - \sinh^2(x) &= 1 \\[0.5em] 1 - \tanh^2(x) &= \textrm{sech}^2(x) \\[0.5em] \coth^2(x) - 1 &= \textrm{csch}^2(x) \end{aligned}\]
Sum and difference formulas
For all real numbers \(x,y\):
\[\begin{aligned} \cosh(x + y) &= \cosh(x)\cosh(y) + \sinh(x)\sinh(y) \\[0.5em] \cosh(x - y) &= \cosh(x)\cosh(y) - \sinh(x)\sinh(y) \\[0.5em] \sinh(x + y) &= \sinh(x)\cosh(y) + \cosh(x)\sinh(y) \\[0.5em] \sinh(x - y) &= \sinh(x)\cosh(y) - \cosh(x)\sinh(y) \\[0.5em] \tanh(x + y) &= \frac{\tanh(x) + \tanh(y)}{1 + \tanh(x)\tanh(y)}\\[0.5em] \tanh(x - y) &= \frac{\tanh(x) - \tanh(y)}{1 - \tanh(x)\tanh(y)} \end{aligned}\]
Double angle formulas
For all real numbers \(x\):
\[\begin{aligned} \cosh(2x) &= \cosh^2(x) + \sinh^2(x) \\[0.5em] \sinh(2x) &= 2\sinh(x)\cosh(x) \\[0.5em] \tanh(2x) &= \frac{2\tanh(x)}{1 + \tanh^2(x)} \end{aligned}\]
Definitions of inverse hyperbolic functions
| function | logarithmic definition | validity |
|---|---|---|
| \(\sinh^{-1}(x)\) | \(\ln\left(x + \sqrt{x^2 + 1}\right)\) | |
| \(\cosh^{-1}(x)\) | \(\ln\left(x + \sqrt{x^2 - 1}\right)\) | \(x\geq 1\) |
| \(\tanh^{-1}(x)\) | \(\dfrac{1}{2}\ln\left(\dfrac{1+x}{1-x}\right)\) | \(|x| < 1\) |
| \(\textrm{coth}^{-1}(x)\) | \(\dfrac{1}{2}\ln\left(\dfrac{x+1}{x-1}\right)\) | \(|x| > 1\) |
| \(\textrm{sech}^{-1}(x)\) | \(\ln\left(\dfrac{1}{x} + \sqrt{\dfrac{1}{x^2}-1}\right)\) | \(0< x \leq 1\) |
| \(\textrm{csch}^{-1}(x)\) | \(\ln\left(\dfrac{1}{x} + \sqrt{\dfrac{1}{x^2} +1}\right)\) | \(x\neq 0\) |
Version history
v1.0: created in 08/25 by tdhc.