Example 1 Let $P$ be a point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $F_{1}, F_{2}$ be the two foci of the ellipse, and $e$ be the eccentricity of the ellipse. If $\angle P F_{1} F_{2}=\alpha, \angle P F_{2} F_{1}=\beta$, try to express $\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2}$ in terms of $e$.