How to prove that
$$ \int\_{t\_0}^{t\_0+T} \sin(m\omega t)\sin(n\omega t)\,\mathrm{d}t$$ 
will equal to $0$ when $m\ne n$ and $\frac{T}{2}$ when $m=n\ne 0$?
Besides
$$ \int\_{t\_0}^{t\_0+T} \cos(m\omega t)\cos(n\omega t)\,\mathrm{d}t$$
will equal to $0$ when $m\ne n$ and $\frac{T}{2}$ when $m=n\ne 0$ and $T$ when $m=n=0$?