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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf SOLUTIONS -- SET 9}\\%
\vspace {3mm} (February 25, 2009) \\
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\begin{description}

\item{(41.1)} 
$$ \frac{1}{\sqrt{1-\tanh^2(\zeta)}} = \sqrt{\frac{\cosh^2(\zeta)}
   {\cosh^2(\zeta)-\sinh^2(\zeta)}} = \cosh(\zeta)\ .  $$
\item{(41.2)} 
$$\cosh(\zeta_1)\,\cosh(\zeta_2)+\sinh(\zeta_1)\,\sinh(\zeta_2)$$
$$=\frac{e^{\zeta_1}+e^{-\zeta_1}}{2}\,\frac{e^{\zeta_2}+e^{-\zeta_2}}{2} 
+ \frac{e^{\zeta_1}-e^{-\zeta_1}}{2}\,\frac{e^{\zeta_2}-e^{-\zeta_2}}{2}$$
$$=\frac{e^{\zeta_1+\zeta_2}+2\,e^{\zeta_1-\zeta_2}+e^{-\zeta_1-\zeta_2}}{4} 
+ \frac{e^{\zeta_1+\zeta_2}-2\,e^{\zeta_1-\zeta_2}+e^{-\zeta_1-\zeta_2}}{4}$$
$$  = \cosh(\zeta_1+\zeta_2) $$
\item{(41.3)} 
\begin{eqnarray} \nonumber
  \tanh(\zeta_1+\zeta_2) &=& \frac{\sinh(\zeta_1+\zeta_2)}
                                  {\cosh{\zeta_1+\zeta_2}} \\ \nonumber
&=&\frac{\sinh(\zeta_1)\,\cosh(\zeta_2)+\cosh(\zeta_1)\,\sinh(\zeta_2)}
        {\cosh(\zeta_1)\,\cosh(\zeta_2)+\sinh(\zeta_1)\,\sinh(\zeta_2)} 
\\ \nonumber &=&\frac{\tanh(\zeta_1)+\tanh(\zeta_2)}
                     {1+\tanh(\zeta_1)\,\tanh(\zeta_2)} \ =\ 
             \frac{\beta_1+\beta_2}{1+\beta_1\,\beta_2}
\end{eqnarray}

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% \caption{Ladder for assignment~14.} \label{fig_lader} \end{figure}

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