It's easy to see that is n is an odd positive integer, $\displaystyle \sin\Big(\frac{n \pi}{4}\Big) = \sin\Big(\frac{3n \pi}{4}\Big) $

and if n is an even positive integer, $\displaystyle \sin\Big(\frac{n \pi}{4}\Big) = -\sin\Big(\frac{3n \pi}{4}\Big) $

But how would you prove that?