How is $\cos (\pi/2+h)$ equal to $-\sin(h)$?

Question

I am not able to grasp the logic behind how $\cos(\frac \pi2+h) = -\sin(h)$. I was able to find an explanation on Reddit but it is not clear. Can anybody elaborate in a better way?

Here is the link to the explanation - reddit link

Answer 1

Use that $$\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$$

Answer 2

Hint: One way to define $\cos x$ is $$\frac{e^{ix}+e^{-ix}}{2}$$ and, similarly, one can write $\sin x$ as $$\frac{e^{ix}-e^{-ix}}{2i},$$ where $i$ is defined such that $i^2=-1$. Use $e^{i\pi}=-1$.