A combinatorial approach to the Fourier expansions of powers of cos and sin

We present a new combinatorial approach to the computation of the (real) Fourier expansions of $\cos^n(t)$ and $\sin^n(t)$, where $n\geq 1$ is an integer. As an application, we compute the Fourier expansions of $f(t)=\frac{1}{a-\cos t}$ and $g(t)=\frac{1}{a-\sin t}$, where $a\in\mathbb R$ with $|a|>1$.