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:
 * $\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$

or:

Proof
From the discussion of Sine and Cosine are Periodic on Reals:
 * $\map \sin {x + \dfrac \pi 2} = \cos x$

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