I was asked to find range of
$$f(x)=\cos^2x+4\sec^2x$$
I converted the given expression to-
$$f(x)=\cos^2x+\frac{4}{\cos^2x}$$
and then applied AM-GM inequality but when I checked whether equality will hold or not I found that for equality $\cos x=\pm\sqrt{2}$ which is not possible, therefore I can't find range of expression from here.
But when I repeated the same procedure by writing $f(x)=4\sec^2x+\frac{1}{\sec^2x}$. I found that inequality condition is also satisfied and I can easily write range.
I want to know why I can't get range from my first try?
Best Answer
The inequality condition is still not satisfied. $\sec x\neq \pm \frac {1}{\sqrt 2}$ as range of $\sec x$ is $(-\infty, -1] \cup [1, \infty)$.
My advice would be to write the expression as $(\cos x+ 2\sec x)^2-4$ and try to proceed from there.