Proving the inequality $\ln(\cos x)\ge \frac{-x^2}{\cos^2(x)}$ for $x\in [0,\frac{\pi}{2}]$ I want to prove the following inequality: $\ln(\cos x)\ge \dfrac{-x^2}{\cos^2(x)}$
Please I'm stuck with this problem, maybe considering the equivalent inequality $\ln(\sec x)\ge (x \sec x)^2$ would help, but I'm not sure.
I want to prove it by the use of the Mean Value theorem.

Consider $f(x) = x^2 + \cos^2(x) \ln(\cos(x))$. Show that $f(x)$ is increasing for $x \in [0,\pi/2)$ and conclude what you want.
