uniqueness of Fourier expansion

If a real function $f$, Riemann integrable on the interval  $[-\pi ,\pi ]$,  may be expressed as sum of a trigonometric series

$f(x)={\displaystyle \frac{a_0}{2}}+{\displaystyle \sum _{m=1}^{\mathrm{\infty }}}(a_m\cos mx+b_m\sin mx)$(1)

where the series $a_1+b_1+a_2+b_2+a_3+b_3+\mathrm{\dots }$ of the coefficients converges absolutely, then the series (1) converges uniformly on the interval and can be integrated termwise (http://planetmath.org/SumFunctionOfSeries).  The same concerns apparently the series which are gotten by multiplying the equation (1) by $\cos nx$ and by $\sin nx$;  the results of the integrations determine for the coefficients $a_n$ and $b_n$ the unique values

$a_n$$={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\cos nxdx,$

$b_n$$={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_{-\pi }^{\pi }}f(x)\sin nxdx$

for any $n$.  So the Fourier series of $f$ is unique.

As a consequence, we can infer that the well-known goniometric formula

$${\sin }^{2}x=\frac{1-\cos 2x}{2}$$

presents the Fourier series of the even function ${\sin }^{2}x$.