Example 2 The height of a regular quadrilateral pyramid is $h$, and the dihedral angle between two adjacent side faces is $2 \alpha$. Find the volume of this regular quadrilateral pyramid.
Slightly solved: Let the angle between the side face and the base be $\theta$, and the side length of the base of the regular quadrilateral pyramid be $a$. From (*) and the problem statement, we have:
$$
\left\{\begin{array}{l}
\cos ^{2} \theta=-\cos 2 \alpha, \\
\frac{a}{2}=h \operatorname{ctg} \theta .
\end{array}\right.
$$
Squaring both sides of (2) and rearranging, we get:
$$
\begin{aligned}
z^{2} & =4 h^{2} \operatorname{ctg}^{2} \theta \\
& =-2 h^{2} \cos 2 \alpha \sec ^{2} \alpha .
\end{aligned}
$$

Therefore, the volume $V$ of the regular quadrilateral pyramid $=\frac{1}{3} s h$
$$
=-\frac{2}{3} h^{3} \cos 2 \alpha \sec ^{2} \alpha
$$

Using (*) formula, we can also easily find the other two angles from the known $\beta$, $\alpha$, or $\theta$, which is not detailed here.