Sign of Haversine

Theorem
The haversine is non-negative for all $\theta \in \R$.

Proof
The haversine is conventionally defined on the real numbers only.

We have that:
 * $\forall \theta \in \R: -1 < \cos \theta < 1$

and so:
 * $\forall \theta \in \R: 0 < 1 - \cos \theta < 2$

from which the result follows by definition of haversine.