$g_T(t)$ is a function which is equal to zero outside the interval $[0, T]$. $f_c=n/T$, where $n$ is a positive integer. Under what condition, $g_T(t)\cos(2\pi f_c t)$ and $g_T(t)\sin(2\pi f_c t)$ are orthogonal?
Note: We say that $g_T(t)\cos(2\pi f_c t)$ and $g_T(t)\sin(2\pi f_c t)$ are orthogonal if $$ \int_0^T g_T^2(t) \cos(2\pi f_c t) \sin(2\pi f_c t) dt = 0. $$
Since $$ \begin{align} \int_0^Tg_T^2(t)\cos(2\pi f_ct)\sin(2\pi f_ct)\,\mathrm{d}t &=\frac12\int_0^Tg_T^2(t)\sin(4\pi nt/T)\,\mathrm{d}t \end{align} $$ Therefore, if the $2n^{\text{th}}$ $\sin$ coefficient of the Fourier Series for $g_T^2$ is $0$, the integral vanishes.