%This command provides the text for the hyperbolic function in
%the second part of the second column in page 4
%
%This macro has one parameter
%      1) The width allowed to typeset the text
\newcommand\TFourHyperbolic[1]{%
   \parbox[t]{#1}{%
      \DisplaySpace{\TFourDisplaySpace}{\TFourDisplayShortSpace}

      \TFourTitle{Definitions:}
       \begin{DisplayFormulae}{1}{0pt}{\TFourSkipFormulae}{\BigChar}{\StyleWithoutNumber}
        \def\LineOfArray##1##2##3##4{\rule{0pt}{4ex plus .5ex minus 1ex}%
                                     {##1}&=&{##2}&{##3}&=&{##4}\\}
        \Fm{\begin{array}{l%Equal sign
                          @{\hspace{.1em}}c@{\hspace{.2em}}%
                          l%
                          l%
                          @{\hspace{.1em}}c@{\hspace{.2em}}%
                          l}
                 \LineOfArray{\sinh x}{\frac{e^x - e^{-x}}{2}}%
                             {\cosh x}{\frac{e^x + e^{-x}}{2}}
                 \LineOfArray{\tanh x}{\frac{e^x - e^{-x}}{e^x + e^{-x}}}
                             {\csch x}{\frac{1}{\sinh x}}
                 \LineOfArray{\sech x}{\frac{1}{\cosh x}}
                             {\coth x}{\frac{1}{\tanh x}}
              \end{array}%
           }
      \end{DisplayFormulae}

      \TFourTitle{Identities:}
      \AdjustSpace{1ex plus .5ex minus .5ex}
      \begin{DisplayFormulae}{1}{0pt}{\TFourSkipFormulae}{\BigChar}{\StyleWithoutNumber}
          \def\FmSep{\unskip\text{,}}%use a comma as separator for the formula
          \Fm{\cosh^2 x - \sinh^2 x = 1}
          \Fm{\tanh^2 x + \sech^2 x = 1}
          \Fm{\coth^2 x - \csch^2 x = 1}
          \Fm{\sinh (-x) = - \sinh x}
          \Fm{\cosh (-x) = \cosh x}
          \Fm{\tanh (-x) = - \tanh x}
          \Fm{\sinh (x + y) = \sinh x \cosh y + \cosh x \sinh y}
          \Fm{\cosh (x + y) = \cosh x \cosh y + \sinh x \sinh y}
          \Fm{\sinh 2x = 2 \sinh x \cosh x}
          \Fm{\cosh 2x = \cosh^2 x + \sinh^2 x}
          \Fm{\cosh x + \sinh x = e^x}
          \Fm{\cosh x - \sinh x = e^{-x}}
          \def\FirstPart{(\cosh x + \sinh x)^n = \mbox{}}
          \FmPartA{\FirstPart\cosh n x +}
          \FmPartB{\FirstPart}{\sinh n x \MathRemark{n \in \Z}}
          \Fm{2 \sinh^2 \tfrac{x}{2} = \cosh x - 1}
          \def\FmSep{\unskip\text{.}}
          \Fm{2 \cosh^2 \tfrac{x}{2} = \cosh x + 1}
      \end{DisplayFormulae}
   }
}
