Why can $$\sum_{n=1}^N a_n \frac{e^{2 \pi i n t} + e^{-2 \pi i n t}}{2} + b_n \frac{e^{2 \pi i n t} - e^{-2 \pi i n t}}{2i} $$ be written as $$ \sum_{n=-N}^N c_n e^{2 \pi i n t}$$ for some setting of the coefficients $c_n$?
My complex math is a bit rusty, and I'm having trouble seeing how the $c_n$s should be set. I know that we will somehow end up with $c_{-n} = \overline{c_n}$. I have tried breaking it down to $sin$s and $cos$, but that seems to be going in the opposite direction.