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**fardeen_gen** Prove that if $\displaystyle m, n\ \mbox{and}\ p$ are arbitrary integers then the expression:

$\displaystyle \sin \frac{m\pi}{p}\sin \frac{n\pi}{p}\ +\ \sin \frac{2m\pi}{p}\sin \frac{2n\pi}{p}\ +\ \sin \frac{3m\pi}{p}\sin \frac{3n\pi}{p}\ +$ $\displaystyle \mbox{...}\ +\ \sin \frac{(p - 1)m\pi}{p}\sin \frac{(p - 1)n\pi}{p}$ is equal to:

$\displaystyle \blacksquare\ -\frac{p}{2}$ when $\displaystyle (m + n)$ is divisible by $\displaystyle 2p$ and $\displaystyle (m - n)$ is not,

$\displaystyle \blacksquare\ \frac{p}{2}$ when $\displaystyle (m - n)$ is divisible by $\displaystyle 2p$ and $\displaystyle (m + n)$ is not,

$\displaystyle \blacksquare\ 0$ when both $\displaystyle (m + n)$ and $\displaystyle (m - n)$ are divisible by $\displaystyle 2p$ or not divisible by $\displaystyle 2p$.