Question 29, Given $\theta \in\left[0, \frac{\pi}{2}\right]$, if $\theta$ satisfies the inequality $\sin ^{3} \theta-\cos ^{3} \theta \geq \ln \cot \theta$, find the range of values for $\theta$.
Question 29, Solution: In fact,
$$
\begin{array}{l}
\sin ^{3} \theta-\cos ^{3} \theta \geq \ln \cot \theta \\
\Leftrightarrow \sin ^{3} \theta-\cos ^{3} \theta \geq \ln \cos \theta-\ln \sin \theta \\
\Leftrightarrow \sin ^{3} \theta+\ln \sin \theta \geq \cos ^{3} \theta+\ln \cos \theta
\end{array}
$$

Let $f(x)=x^{3}+\ln x$, it is clear that $f(x)$ is monotonically increasing, hence $\sin \theta \geq \cos \theta$, thus $\theta \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)$.