$∃$ no $c$ s.t. $f\:'\left(x\right)=0\ \rightarrow \left(\frac{1}{cos^2x}\right)=0\:\forall \:c\subset \:x\:\in \:\left]0,\pi \right[$
I have confusion in the latter part- what I infer from it is, $c$ are the specific $x$ (thus being a subset of $x$) in the open interval of $\left(0,\pi \right)$. Am I right?
Yes, you're right.
The statement is saying that $c$ is some arbitrary subset of $x$
$x$ is in the interval $(0, \pi)$
So the first statement says that there exists no set of values $c$ which is a subset of $x$ such that...