# 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.

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