Dirichlet's theorem says that any function $f(x)$ on the interval $[-a,+a]$ can be expanded as a Fourier series: $$f\left ( x \right )=\sum_{n=0}^{\infty}\left [ a_{n} \sin \left ( \frac{n\pi x}{a}\right)+b_{n}\cos\left ( \frac{n \pi x}{a} \right ) \right ].$$ Show that this can be written equivalently as $$f\left ( x \right )=\sum_{n=-\infty}^{\infty}c_{n}\exp\left\{\frac{i n \pi x}{a}\right\}.$$
Here's what there is thus far
expressing the sin and cos as complex exponentials I arrive at
$$a_{n}\sin\left ( \frac{n \pi x}{a} \right )+b_{n}\cos\left ( \frac{n \pi x}{a} \right )=e^{i\left ( \frac{n \pi x}{a} \right )}\left [ \frac{-a_{n}i}{2} +\frac{b_{n}}{2}\right ]+e^{-i\left ( \frac{n \pi x}{a} \right )}\left [ \frac{a_{n}i}{2}+\frac{b_{n}}{2} \right ].$$
I wonder if I'm being confused with the notations.
\begin{align} &\frac{1}{a}\int_{-a}^{a}f(t)\sin(n\pi t/a)dt\sin(n\pi x/a)+\frac{1}{a}\int_{-a}^{a}f(t)\cos(n\pi t/a)dt\cos(n\pi x/a) \\ & = \frac{1}{a}\int_{-a}^{a}f(t)\cos(n\pi(x-t)/a)dt \\ & = \frac{1}{a}\int_{-a}^{a}f(t)\frac{e^{in\pi(x-t)/a}+e^{-in\pi(x-t)}}{2}dt \\ & = \frac{1}{2a}\int_{-a}^{a}f(t)e^{-in\pi t/a}dt e^{in\pi x/a} + \frac{1}{2a}\int_{-a}^{a}f(t)e^{in\pi t/a}dt e^{-in\pi x/a}. \end{align}