Fourier cosine coefficients of a sine mode ($a_n=4m/[\pi (m^2-n^2)]\text{ when }m,n\text{ have opposite parity}$)

For a positive integer $m$, take $f(x)=\sin (mx)$ on $0<x<\pi$ and form its half-range Fourier cosine series. The product-to-sum identity givesThus $a_n=4m/[\pi (m^2-n^2)]$ for opposite parity and zero for the same parity. The exceptional integral $a_m=(2/\pi ){\int }_0^{\pi }\sin (mx)\cos (mx)dx$ is zero. The constant term is $a_0/2$, not $a_0$. These coefficients represent the even periodic extension, not the original sine wave on negative arguments. Their quadratic decay gives absolute convergence.