Finding the Fourier series of $f(x)=\sin x \cos^2x$

Question

Let us have $f(x)=\sin x \cos^2x$. We need to get the Fourier-series of this.

Should we make $f(x)$ nicer using the known identities between $\sin, \cos$? I tried using, that $\cos^2x=\frac12 + \frac12 \cos(2x)$.

How should I move forward, and what is the general approach for this kind of tasks? :)

Any help appreciated.

Answer 1

Hint. One may write $$ f(x)=\sin x \cos^2x=\frac12\sin(2x)\cdot\cos x=\frac14\left(\sin(3x)+\sin(x)\right) $$ which is clearly a Fourier series over $[0,2\pi]$.