%%% ================================================================
%\newpage
%\section{Trigonometric identities}
%
%Proving orthonormalcy of the $e^{i2\pi k x}$'s on the unit interval is a breeze
%--- one of the benefits of using complex numbers.  Proving orthonormalcy of the
%$\sin(2\pi kx)$'s and $\cos(2\pi kx)$'s is less immediate, but also easy given
%the correct trig identities.
%
%The fundamental identities are the Pythagorean identity and the sum and
%difference formulas:
%\begin{align*}
%	\sin^2(\alpha) + \cos^2(\alpha) &= 1
%\end{align*}
%\begin{align*}
%	\sin(\alpha + \beta) &= \sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)\\
%	\sin(\alpha - \beta) &= \sin(\alpha)\cos(\beta)-\cos(\alpha)\sin(\beta)\\
%	\cos(\alpha + \beta) &= \cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)\\
%	\cos(\alpha - \beta) &= \cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta).
%\end{align*}
%Adding and subtracting pairs of equations gives us the following.  Note that
%they turn the \emph{products} which appear in the orthonormalcy formulas into
%\emph{sums} which are easy to integrate over:
%\begin{align*}
%	\sin(\alpha)\cos(\beta)
%	&= \frac{1}{2} \left(\sin(\alpha + \beta) + \sin(\alpha - \beta)\right)\\
%	\sin(\alpha)\sin(\beta)
%	&= \frac{1}{2} \left(\cos(\alpha - \beta) - \cos(\alpha + \beta)\right)\\
%	\cos(\alpha)\cos(\beta)
%	&= \frac{1}{2} \left(\cos(\alpha + \beta) + \cos(\alpha - \beta)\right).
%\end{align*}
