Sine of Half-Integer Multiple of Pi

Theorem
Let $x \in \R$ be a real number.

Let $\sin x$ be the sine of $x$.

Then:
 * $\forall n \in \Z: \sin \left({n + \dfrac 1 2}\right) \pi = \left({-1}\right)^n$

or alternatively:

Proof
From the discussion of Sine and Cosine are Periodic on Reals, we have that:
 * $\sin \left({x + \dfrac \pi 2}\right) = \cos x$

The result then follows directly from the Cosine of Multiple of Pi.