<s>[INST] 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.
 [/INST] $$
\ln (1+x) = x - \frac{x^2}2 + \frac{x^3}3 - \cdots.
$$
$$
\ln\cos x = \ln\left(1 -\frac{x^2}{2} + \frac{x^4}{24}-\cdots\right) \ge\ln\left(1 -\frac{x^2}{2}\right)
$$
$$
=\frac{-x^2}{2} -\frac{x^4}{8} + \cdots \ge -\frac{x^2}{1.5} \ge-\frac{x^2}{\cos^2 x}\text{ for $x$ close enough to $0$.}
$$
</s>