# Hyperbolic Functions Formulas and Identities

 $\DeclareMathOperator{\sech}{sech}$ $\DeclareMathOperator{\csch}{csch}$

## Definitions of Hyperbolic Functions

• Hyperbolic sine : $\sinh x = \dfrac{e^x - e^{-x}}{2}$
• Hyperbolic cosine : $\cosh x = \dfrac{e^x + e^{-x}}{2}$
• Hyperbolic tangent : $\tanh x = \dfrac{\sinh x}{\cosh x}$
• Hyperbolic cotangent : $\coth x = \dfrac{1}{\tanh x} = \dfrac{\cosh x}{\sinh x}$
• Hyperbolic secant : $\sech x = \dfrac{1}{\cosh x}$
• Hyperbolic cosecant : $\csch x = \dfrac{1}{\sinh x}$

## Negative Argument Formulas

• $\sinh (-x) = - \sinh x$ , odd function , graph symmetric with respect to the origin.
• $\cosh (-x) = \cosh x$ , even function , graph symmetric with $y$ axis.
• $\tanh (-x) = - \tanh x$ , odd function , graph symmetric with respect to the origin.

## Hyperbolic Functions of Sums/Difference

• $\cosh(A \pm B) = \cosh A \cosh B \pm \sinh A \sinh B$
• $\sinh(A \pm B) = \sinh A \cosh B \pm \cosh A \sinh B$
• $\tanh(A \pm B) = \dfrac{\tanh A \pm \tanh B}{1 \pm \tanh A \tanh B}$

## Sums of Hyperbolic Functions to Product of Hyperbolic Functions

• $\sinh A + \sinh B = 2 \sinh \left( \dfrac {A+B}{2} \right) \cosh \left( \dfrac {A-B}{2} \right)$
• $\sinh A - \sinh B = 2 \sinh \left( \dfrac {A - B}{2} \right) \cosh \left( \dfrac {A + B}{2} \right)$
• $\cosh A + \cosh B = 2 \cosh \left( \dfrac {A+B}{2} \right) \cosh \left( \dfrac {A-B}{2} \right)$
• $\cosh A - \cosh B = 2 \sinh \left( \dfrac {A+B}{2} \right) \sinh \left( \dfrac {A-B}{2} \right)$

## Product of Hyperbolic Functions to Sum of Hyperbolic Functions

• $\sinh A \cosh B = \dfrac{1}{2} \; [ \sinh(A+B) + \sinh(A-B) ]$
• $\cosh A \cosh B = \dfrac{1}{2} \; [ \cosh(A+B) + \cosh(A-B) ]$
• $\sinh A \sinh B = \dfrac{1}{2} \; [ \cosh(A+B) - \cosh(A-B) ]$

## Hyperbolic Functions of Multiple Arguments

• $\sinh (2 A) = 2 \sinh A \cosh A$
• $\cosh (2 A) = 2 \sinh^2 A + 1 = 2 \cosh^2 A -1$
• $\sinh (3 A) = 4 \sinh^3 A + 3 \sinh A$
• $\cosh (3 A) = 4\cosh^3 A - 3 \cosh A$

## Relations Between Hyperbolic Functions

• $\cosh^2 A - \sinh^2 A = 1$
• $\tanh^2 A + \sech^2 A = 1$
• $\coth^2 A - \csch^2 A = 1$

## Power Reducing Identities

• $\sinh^2 A = \dfrac{1}{2} [\cosh (2A) - 1]$
• $\cosh^2 A = \dfrac{1}{2} [\cosh (2A) + 1]$
• $\sinh^3 A = \dfrac{\sinh(3A) - 3 \sinh A}{4}$
• $\cosh^3 A = \dfrac{\cosh(3A) + 3 \cosh A}{4}$
• $\sinh^4 A = \dfrac{\cosh (4A) - 4 \cosh (2A) + 3}{8}$
• $\cosh^4 A = \dfrac{\cosh (4A) + 4 \cosh (2A) + 3}{8}$
• $\sinh^5 A = \dfrac{\sinh (5A) - 5 \sinh (3A) + 10 \sinh A}{16}$
• $\cosh^5 A = \dfrac{\cosh (5A) + 5 \cosh (3A) + 10 \cosh A}{16}$

## Inverse Hyperbolic Functions

• $\sinh^{-1} A = \ln (x+\sqrt{x^2+1})$
• $\cosh^{-1} A = \ln (x+\sqrt{x^2-1}) \; , \; x \ge 1$
• $\tanh^{-1} A = \dfrac{1}{2}\ln \left(\dfrac{1+x}{1-x} \right) \; , \; -1 \lt x \lt 1$
• $\csch^{-1} A = \ln \left(\dfrac{1}{x}+\sqrt{\dfrac{1}{x^2} + 1} \right) \; , \; x \ne 0$
• $\sech^{-1} A = \ln \left(\dfrac{1}{x}+\sqrt{\dfrac{1}{x^2} - 1} \right) \; , \; 0 \lt x \le 1$
• $\coth^{-1} A = \dfrac{1}{2}\ln \left(\dfrac{x+1}{x-1} \right) \; , \; |x| \gt 1$

## Derivative of Hyperbolic Functions and Their Inverses

### $\dfrac{d f(x)}{dx}$

$\sinh x$ $\cosh x$
$\cosh x$ $\sinh x$
$\tanh x$ $\text{sech}^2 x$
$\coth x$ $- \text{csch}^2 x$
$\text{sech} \; x$ $-\text{sech} \; x \tanh x$
$\text{csch} \; x$ $- \text{csch} \; x \coth x$
$\sinh^{-1} x$ $\dfrac{1}{\sqrt{x^2+1}}$
$\cosh^{-1} x$ $\dfrac{1}{\sqrt{x^2-1}}$
$\tanh^{-1} x$ $\dfrac{1}{1-x^2}$
$\coth^{-1} x$ $\dfrac{1}{1-x^2}$

## Indefinite Integrals of Hyperbolic Functions

The constant of integration is omitted here but should be added if necessary.

### $\displaystyle \int f(x) dx$

$\sinh x$ $\cosh x$
$\cosh x$ $\sinh x$
$\tanh x$ $\ln( \cosh x)$
$\coth x$ $\ln( \sinh x)$
$\text{sech} \; x$ $2 \tan^{-1}(e^x)$
$\text{csch} \; x$ $-\ln (\coth x + \text{csch}\; x)$