I understand that the max value of $1 / \sec x = 1$ mathematically.
However graphically, I don't understand it. To get the max value of 1 / sec x, sec x has to be the minimum value which is undefined on the graph. So can someone explain how to solve this graphically?
Consider the graph of the function $f(x) = 1/x$, which maps large positive values to tiny positive values (and vice versa), but maps large negative values to tiny negative values (and vice versa) as well. Note further that $f(1)=1$ and $f(-1)=-1$.
From that graph you now can graphically deduce $g(x) = (f\circ\sec )(x) = 1/\sec(x)$: Obviously you would have $g(\pi\cdot 2k)=+1$, $g(\pi/2\cdot(2k+1))=0$, $g(\pi\cdot(2k+1))=-1$, each for any integer $k$. Moreover you get that $y=g(x)$ would oscilate endlessly between $+1$ and $-1$.
(In fact, since $\sec(x):=1/\cos(x)$ you would get $g(x) = 1/\sec(x)=\cos(x)$.)
However already the first graphical consideration shows that $y=g(x)$ surely has not only a well-defined maximum ($y=+1$), but also a well-defined minimum ($y=-1$).
--- rk