Zeroes of Sine and Cosine/Cosine

Theorem
Let $x \in \R$.


 * $\cos x = 0$, $x = \paren {n + \dfrac 1 2} \pi$ for some $n \in \Z$.

Proof
From Sine and Cosine are Periodic on Reals: Corollary:

$\cos x$ is:
 * strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$

and:
 * strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$

The result follows directly from Sine and Cosine are Periodic on Reals.