Prove $\max \cos(x)$ is $1$ and $\min \cos(x)$ is $ -1$ Prove $\max \cos(x)$ is $1$ and $\min \cos(x)$ is $-1$
How to prove it with only calculus and not multivariable calculus?
Please notice that this is not a homework question, but a pre-exam question. Thanks a lot.

It is $\cos{(\varphi)}=\Re{({e^{i\varphi}})}$ and $|e^{i\varphi}|=1$.
$|\Re{(z)}|\leq|z|$ together with the evaluation at $0$ and $\pi$ proves your question.
