addition and subtraction formulas for hyperbolic functions

The addition formulas for hyperbolic sine, hyperbolic cosine, and hyperbolic tangent will be achieved via brute .

$\sinh (x+y)$ $={\displaystyle \frac{e^{x+y}-e^{-(x+y)}}{2}}$

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

$=e^x\left({\displaystyle \frac{e^y-e^{-y}}{2}}\right)+e^{-y}\left({\displaystyle \frac{e^x-e^{-x}}{2}}\right)$

$=(\cosh x+\sinh x)\sinh y+(\cosh y-\sinh y)\sinh x$

$=\cosh x\sinh y+\sinh x\sinh y+\sinh x\cosh y-\sinh x\sinh y$

$=\sinh x\cosh y+\cosh x\sinh y$

$\cosh (x+y)$ $={\displaystyle \frac{e^{x+y}+e^{-(x+y)}}{2}}$

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

$=e^x\left({\displaystyle \frac{e^y-e^{-y}}{2}}\right)+e^{-y}\left({\displaystyle \frac{e^x+e^{-x}}{2}}\right)$

$=(\cosh x+\sinh x)\sinh y+(\cosh y-\sinh y)\cosh x$

$=\cosh x\sinh y+\sinh x\sinh y+\cosh x\cosh y-\cosh x\sinh y$

$=\cosh x\cosh y+\sinh x\sinh y$

$\tanh (x+y)$ $={\displaystyle \frac{\sinh (x+y)}{\cosh (x+y)}}$

$={\displaystyle \frac{\sinh x\cosh y+\cosh x\sinh y}{\cosh x\cosh y+\sinh x\sinh y}}$

$={\displaystyle \frac{{\displaystyle \frac{\sinh x}{\cosh x}}\cdot {\displaystyle \frac{\cosh y}{\cosh y}}+{\displaystyle \frac{\cosh x}{\cosh x}}\cdot {\displaystyle \frac{\sinh y}{\cosh y}}}{{\displaystyle \frac{\cosh x}{\cosh x}}\cdot {\displaystyle \frac{\cosh y}{\cosh y}}+{\displaystyle \frac{\sinh x}{\cosh x}}\cdot {\displaystyle \frac{\sinh y}{\cosh y}}}}$

$={\displaystyle \frac{\tanh x+\tanh y}{1+\tanh x\tanh y}}$

Note that $\sinh$ and $\tanh$ are odd functions and $\cosh$ is an even function, i.e. (http://planetmath.org/Ie) $\sinh (-t)=-\sinh t$, $\tanh (-t)=-\tanh t$, and $\cosh (-t)=\cosh t$. These facts enable us to obtain the subtraction formulas.

$$\sinh (x-y)=\sinh (x+(-y))=\sinh x\cosh (-y)+\cosh x\sinh (-y)=\sinh x\cosh y-\cosh x\sinh y$$

$$\cosh (x-y)=\cosh (x+(-y))=\cosh x\cosh (-y)+\sinh x\sinh (-y)=\cosh x\cosh y-\sinh x\sinh y$$

$$\tanh (x-y)=\tanh (x+(-y))=\frac{\tanh x+\tanh (-y)}{1+\tanh x\tanh (-y)}=\frac{\tanh x-\tanh y}{1-\tanh x\tanh y}$$

Title	addition and subtraction formulas for hyperbolic functions
Canonical name	AdditionAndSubtractionFormulasForHyperbolicFunctions
Date of creation	2013-03-22 17:50:45
Last modified on	2013-03-22 17:50:45
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	5
Author	Wkbj79 (1863)
Entry type	Derivation
Classification	msc 26A09
Classification	msc 33B10
Synonym	addition and subtraction formulae for hyperbolic functions
Synonym	addition formulas for hyperbolic functions
Synonym	addition formulae for hyperbolic functions
Synonym	subtraction formulas for hyperbolic functions
Synonym	subtraction formulae for hyperbolic functions
Synonym	addition form
Related topic	AdditionFormula
Related topic	HyperbolicIdentities
Related topic	AdditionFormulas