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 $n\ge 1$ 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 $\mathrm{\mid }a\mathrm{\mid }>1$.