Example 2. Two regular quadrilateral pyramids with a common base but opposite directions form an octahedron. If the side length of the base of the regular quadrilateral pyramid is $a$, and the dihedral angle at the vertex is $\alpha$, find the surface area of its inscribed sphere.
$$
\begin{array}{l}
\text{Let the height of the regular quadrilateral pyramid be } h, \text{ the slant height be } h', \text{ and the area of one side face be } S_{1}, \text{ then} \\
h=\frac{a}{2} \sqrt{\cot^{2} \frac{\alpha}{2}}, \\
\dot{n}^{\prime}=\frac{a}{2} \cot \frac{\alpha}{2}, S_{1}=\frac{a^{2}}{4} \cot \frac{\alpha}{2}. \\
\therefore V \text{ of the octahedron }=2 \text{ V of the quadrilateral pyramid }=\frac{a^{3}}{3} \sqrt{\cot^{2} \frac{\alpha}{2}-1}, \\
S \text{ of the octahedron }=8 S_{1}=2 a^{2} \cot \frac{\alpha}{2}, \\
\therefore r=\frac{3 V}{S}=\frac{a}{2} \sqrt{1-\tan^{2} \frac{\alpha}{2}}. \\
\therefore S \text{ of the sphere }=4 \pi r^{2}=4 \pi \cdot \frac{a^{2}}{4}\left(1-\tan^{2} \frac{\alpha}{2}\right) \\
\quad=\frac{2 \pi a^{2} \cos \alpha}{1+\cos \alpha}.
\end{array}
$$