Supremum and maximum

Question

I'm asked to tell whether the supremum and maximum of the following functions exist and to derive them when $x \in B$:

$1- B = [0, 4 \pi)~\text{and}~g(x)=\cos x$

$2- B = [0, 4 \pi)~\text{and}~g(x)=\sin^2x+\cos^2x$

For the first one, I found that $\max f(x)_{x \in B}$ doesn't exist and $\sup f(x)_{x \in B} = -\infty$. Is it correct ?

For the second one, I need help.

Thanks!

Answer 1

Hint: your conclusion for the first one seems absurd, since $\cos x$ takes values in $[-1,1]$. For the second, observe that $\sin^2 x + \cos^2 x = 1$.