Example 4 If
$$
\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right)(\theta \in[0,2 \pi)),
$$

then the range of values for $\theta$ is $\qquad$ [3]
(2011, National High School Mathematics Joint Competition)
【Analysis】The inequality has different degrees on both sides, unlike Example 3 where the function to be constructed can be clearly seen. However, a simple transformation of the inequality will reveal the solution.
The transformation is as follows:
$$
\begin{array}{l}
\cos ^{5} \theta-\sin ^{5} \theta \cos ^{3} \theta+\frac{1}{7} \cos ^{5} \theta .
\end{array}
$$

By comparing the transformed inequality, we can construct the function
$$
f(x)=x^{3}+\frac{1}{7} x^{5} .
$$

The original inequality can be transformed into $f(\sin \theta)>f(\cos \theta)$. Since $f(x)$ is an increasing function on $\mathbf{R}$, it follows that $\sin \theta>\cos \theta$.
Given $\theta \in[0,2 \pi)$, then $\theta \in\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)$.