# 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.

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

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