confused about why this coefficient of fourier series is defined as so

Question

First quesiton solved! The second question about this page is that, I am really confused about why the coefficient $C_n$ is defined like this, quotient of two vectors?

Answer 1

$$\int_{-L}^L \cos((n-m)\pi x/L) dx = \dfrac L {(n-m)\pi}\sin((n-m)\pi x/L)\bigg|_{-L}^{L},\\$$

and $\sin \pm k\pi=0$, where $k=n-m$ is an integer.