Zeroes of Sine and Cosine

Theorem

$(1): \quad \forall n \in \Z: x = \left({n + \dfrac 1 2}\right) \pi \implies \cos x = 0$

$(2): \quad \forall n \in \Z: x = n \pi \implies \sin x = 0$

Proof

From Sine and Cosine are Periodic on Reals: Corollary:

$\cos x$ is:

strictly positive on the interval $\displaystyle \left({-\frac \pi 2 .. \frac \pi 2}\right)$

and:

strictly negative on the interval $\displaystyle \left({\frac \pi 2 .. \frac {3 \pi} 2}\right)$

$\sin x$ is:

strictly positive on the interval $\left({0 .. \pi}\right)$

and:

strictly negative on the interval $\left({\pi .. 2 \pi}\right)$

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

$\blacksquare$