PlanetMath: addition and subtraction formulas for hyperbolic functions

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addition and subtraction formulas for hyperbolic functions

(Derivation)

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

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

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

$\displaystyle \tanh(x+y)$	$\displaystyle =\frac{\sinh(x+y)}{\cosh(x+y)}$
	$\displaystyle =\frac{\sinh x\cosh y+\cosh x\sinh y}{\cosh x\cosh y+\sinh x\sinh y}$
	$\displaystyle =\frac{\displaystyle \frac{\sinh x}{\cosh x} \cdot \frac{\cosh y}... ... \frac{\cosh y}{\cosh y}+\frac{\sinh x}{\cosh x} \cdot \frac{\sinh y}{\cosh y}}$
	$\displaystyle =\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}$

Note that $\sinh$ and $\tanh$ are odd functions and $\cosh$ is an even function, i.e. $\sinh(-t)=-\sinh t$ , $\tanh(-t)=-\tanh t$ , and $\cosh(-t)=\cosh t$ . These facts enable us to obtain the subtraction formulas. $$ \sinh(x-y)=\sinh(x+(-y))=\sinh x\cosh(-y)+\cosh x\sinh(-y)=\sinh x\cosh y-\cosh x\sinh y$$ $$ \cosh(x-y)=\cosh(x+(-y))=\cosh x\cosh(-y)+\sinh x\sinh(-y)=\cosh x\cosh y-\sinh x\sinh y$$ $$ \tanh(x-y)=\tanh(x+(-y))=\frac{\tanh x+\tanh(-y)}{1+\tanh x\tanh(-y)}=\frac{\tanh x-\tanh y}{1-\tanh x\tanh y}$$

"addition and subtraction formulas for hyperbolic functions" is owned by Wkbj79.

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Other names:

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Cross-references: subtraction formulas, even function, odd functions, hyperbolic tangent, hyperbolic cosine, hyperbolic sine, addition formulas
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This is version 2 of addition and subtraction formulas for hyperbolic functions, born on 2008-02-22, modified 2008-02-22.
Object id is 10317, canonical name is AdditionAndSubtractionFormulasForHyperbolicFunctions.
Accessed 6514 times total.

Classification:

AMS MSC:	26A09 (Real functions :: Functions of one variable :: Elementary functions)
	33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)

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