Prove that $2\pi$ periodic function, $f$, with Fourier coefficients equal zero if $n$ is odd

Question

Let $f$ be a continuous, $2\pi$-periodic function and suppose that its Fourier coefficients, $a_n$ and $b_n$, equal $0$ when $n$ is odd, i.e., \begin{equation} \int_{-\pi}^\pi f(x)\cos (nx)\,dx=\int_{-\pi}^\pi f(x)\sin (nx)\,dx=0,\quad n \ne 2k. \end{equation}

Is there a way to prove that $f$ is $\pi$-periodic?