<reponame>tmilev/freecalc<gh_stars>1-10
\begin{frame}
\frametitle{Trig Functions of Differences of Angles}
\begin{example} 
Prove the identities

\hfil \hfil $\begin{array}{rcl}
\sin (\alpha- \beta)&= & \sin\alpha\cos\beta -\cos \alpha\sin \beta \\
\cos(\alpha-\beta)&=&\cos\alpha\cos \beta +\sin\alpha\sin\beta
\end{array}$

from the (already demonstrated) identities

\hfil \hfil $\begin{array}{rcl}
\sin (\alpha+ \beta)&= & \sin\alpha\cos\beta +\cos \alpha \sin \beta \\
\cos(\alpha+\beta)&=&\cos\alpha\cos \beta -\sin\alpha\sin\beta\\
\end{array}
$

$
\begin{array}{rcll|l}
\sin (\alpha-\beta) &=&\sin (\alpha+(-\beta))\\
&=&\sin \alpha \cos (-\beta) +\cos\alpha\sin (-\beta) && \begin{array}{l}\cos \text{ is even },\\ \sin \text{ is odd}\end{array} \\
&=&\sin \alpha \cos \beta -\cos\alpha \sin \beta\\
\cos (\alpha-\beta) &=&\cos (\alpha+(-\beta))\\
&=&\cos \alpha \cos (-\beta) -\sin \alpha\sin(-\beta) && \begin{array}{l}\cos \text{ is even },\\ \sin \text{ is odd}\end{array} \\
&=&\cos \alpha \cos \beta +\cos\alpha \sin \beta
\end{array}
$


\end{example}
\vskip 10cm
\end{frame}
