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addition and subtraction formulas for hyperbolic functions

\documentclass{article}
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\begin{document}
The addition formulas for hyperbolic sine, hyperbolic cosine, and hyperbolic tangent will be achieved via brute \PMlinkescapetext{force}.

\begin{align*}
\sinh(x+y) & =\frac{e^{x+y}-e^{-(x+y)}}{2} \\
& =\frac{e^xe^y-e^xe^{-y}+e^xe^{-y}-e^{-x}e^{-y}}{2} \\
& =e^x\left(\frac{e^y-e^{-y}}{2}\right)+e^{-y}\left(\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
\end{align*}

\begin{align*}
\cosh(x+y) & =\frac{e^{x+y}+e^{-(x+y)}}{2} \\
& =\frac{e^xe^y-e^xe^{-y}+e^xe^{-y}+e^{-x}e^{-y}}{2} \\
& =e^x\left(\frac{e^y-e^{-y}}{2}\right)+e^{-y}\left(\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
\end{align*}

\begin{align*}
\tanh(x+y) & =\frac{\sinh(x+y)}{\cosh(x+y)} \\
& =\frac{\sinh x\cosh y+\cosh x\sinh y}{\cosh x\cosh y+\sinh x\sinh y} \\
& =\frac{\ds\frac{\sinh x}{\cosh x} \cdot \frac{\cosh y}{\cosh y}+\frac{\cosh x}{\cosh x} \cdot \frac{\sinh y}{\cosh y}}
        {\ds\frac{\cosh x}{\cosh x} \cdot \frac{\cosh y}{\cosh y}+\frac{\sinh x}{\cosh x} \cdot \frac{\sinh y}{\cosh y}} \\
& =\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}
\end{align*}

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

\[
\sinh(x-y)=\sinh(x+(-y))=\sinh x\cosh(-y)+\cosh x\sinh(-y)=\sinh x\cosh y-\cosh x\sinh y
\]

\[
\cosh(x-y)=\cosh(x+(-y))=\cosh x\cosh(-y)+\sinh x\sinh(-y)=\cosh x\cosh y-\sinh x\sinh y
\]

\[
\tanh(x-y)=\tanh(x+(-y))=\frac{\tanh x+\tanh(-y)}{1+\tanh x\tanh(-y)}=\frac{\tanh x-\tanh y}{1-\tanh x\tanh y}
\]
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nd{document}