addition and subtraction formulas for hyperbolic functions

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

 $\displaystyle\sinh(x+y)$ $\displaystyle=\frac{e^{x+y}-e^{-(x+y)}}{2}$ $\displaystyle=\frac{e^{x}e^{y}-e^{x}e^{-y}+e^{x}e^{-y}-e^{-x}e^{-y}}{2}$ $\displaystyle=e^{x}\left(\frac{e^{y}-e^{-y}}{2}\right)+e^{-y}\left(\frac{e^{x}% -e^{-x}}{2}\right)$ $\displaystyle=(\cosh x+\sinh x)\sinh y+(\cosh y-\sinh y)\sinh x$ $\displaystyle=\cosh x\sinh y+\sinh x\sinh y+\sinh x\cosh y-\sinh x\sinh y$ $\displaystyle=\sinh x\cosh y+\cosh x\sinh y$
 $\displaystyle\cosh(x+y)$ $\displaystyle=\frac{e^{x+y}+e^{-(x+y)}}{2}$ $\displaystyle=\frac{e^{x}e^{y}-e^{x}e^{-y}+e^{x}e^{-y}+e^{-x}e^{-y}}{2}$ $\displaystyle=e^{x}\left(\frac{e^{y}-e^{-y}}{2}\right)+e^{-y}\left(\frac{e^{x}% +e^{-x}}{2}\right)$ $\displaystyle=(\cosh x+\sinh x)\sinh y+(\cosh y-\sinh y)\cosh x$ $\displaystyle=\cosh x\sinh y+\sinh x\sinh y+\cosh x\cosh y-\cosh x\sinh y$ $\displaystyle=\cosh x\cosh y+\sinh x\sinh y$
 $\displaystyle\tanh(x+y)$ $\displaystyle=\frac{\sinh(x+y)}{\cosh(x+y)}$ $\displaystyle=\frac{\sinh x\cosh y+\cosh x\sinh y}{\cosh x\cosh y+\sinh x\sinh y}$ $\displaystyle=\frac{\displaystyle\frac{\sinh x}{\cosh x}\cdot\frac{\cosh y}{% \cosh y}+\frac{\cosh x}{\cosh x}\cdot\frac{\sinh y}{\cosh y}}{\displaystyle% \frac{\cosh x}{\cosh x}\cdot\frac{\cosh y}{\cosh y}+\frac{\sinh x}{\cosh x}% \cdot\frac{\sinh y}{\cosh y}}$ $\displaystyle=\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}$

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