# hyperbolic identities

There are many formulas involving hyperbolic functions, many of which are to formulas for trigonometric functions. Below is a list of some of these formulas (usually for real arguments).

1. 1.

Hyperbolic version of Pythagorean identities

• $\cosh^{2}x-\sinh^{2}x=1$

• $1-\tanh^{2}x=\operatorname{sech}^{2}x$

• $\coth^{2}x-1=\operatorname{csch}^{2}x$

2. 2.

Fractional identities

• $\displaystyle\tanh x=\frac{\sinh x}{\cosh x}$

• $\displaystyle\coth x=\frac{\cosh x}{\sinh x}$

• $\displaystyle\coth x=\frac{1}{\tanh x}$

• $\displaystyle\tanh x=\frac{1}{\coth x}$

• $\displaystyle\operatorname{csch}x=\frac{1}{\sinh x}$

• $\displaystyle\operatorname{sech}x=\frac{1}{\cosh x}$

3. 3.

Hyperbolic functions of a purely imaginary number

• $\sinh(ix)=i\sin x$

• $\cosh(ix)=\cos x$

• $\tanh(ix)=i\tan x$

• $\cosh(ix)=i\cot x$

• $\operatorname{csch}(ix)=i\csc x$

• $\operatorname{sech}(ix)=\sec x$

4. 4.

• $\sinh(x\pm y)=\sinh x\cosh y\pm\cosh x\sinh y$

• $\cosh(x\pm y)=\cosh x\cosh y\pm\sinh x\sinh y$

• $\displaystyle\tanh(x\pm y)=\frac{\tanh x\pm\tanh y}{1\pm\tanh x\tanh y}$

5. 5.

Formulas for hyperbolic functions of a complex number

• $\sinh(x+iy)=\sinh x\cos y+i\cosh x\sin y$

• $\cosh(x+iy)=\cosh x\cos y+i\sinh x\sin y$

• $\displaystyle\tanh(x+iy)=\frac{\tanh x+i\tan y}{1+i\tanh x\tan y}$

6. 6.

Opposite formulas

• $\sinh(-x)=-\sinh x$

• $\cosh(-x)=\cosh x$

• $\tanh(-x)=-\tanh x$

7. 7.

Double argument formulas

• $\sinh(2x)=2\sinh x\cosh x$

• $\cosh(2x)=\cosh^{2}x+\sinh^{2}x=2\cosh^{2}x-1=1+2\sinh^{2}x$

• $\displaystyle\tanh(2x)=\frac{2\tanh x}{1+\tanh^{2}x}$

8. 8.

Periodicity (http://planetmath.org/Periodic) formulas

• $\sinh(z+2\pi i)=\sinh{z}$

• $\cosh(z+2\pi i)=\cosh{z}$

• $\tanh(z+\pi i)=\tanh{z}$

Cf (http://planetmath.org/Cf). the periodicity of exponential function.

9. 9.

Exponential formulas (http://planetmath.org/ExponentialFunction)

• $\displaystyle\cosh x=\frac{e^{x}+e^{-x}}{2}$

• $\displaystyle\sinh x=\frac{e^{x}-e^{-x}}{2}$

• $\displaystyle\tanh x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$

• $e^{x}=\cosh x+\sinh x$

• $e^{-x}=\cosh x-\sinh x$

Note that the first three formulas given in this are definitions.

 Title hyperbolic identities Canonical name HyperbolicIdentities Date of creation 2013-03-22 17:50:42 Last modified on 2013-03-22 17:50:42 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 11 Author Wkbj79 (1863) Entry type Topic Classification msc 33B10 Classification msc 26A09 Synonym hyperbolic formulas Synonym hyperbolic formulae Related topic GoniometricFormulae Related topic AdditionAndSubtractionFormulasForHyperbolicFunctions Related topic ExamplesOfPeriodicFunctions Related topic TaylorSeriesOfHyperbolicFunctions