Example 10. Suppose that the sum of $a$ and $\beta$ is a constant as they vary. Find the maximum and minimum values of $\sin \alpha \cdot \sin \beta$.
Let $\alpha+\beta=\mathbf{A}$ (a constant), and assume $\alpha=\frac{\mathrm{A}}{2}+\theta$, then $\beta=\frac{\mathbf{A}}{2}-\theta$.
$$
\begin{aligned}
\sin \alpha \cdot \sin \beta & =\sin \left(\frac{A}{2}+\theta\right) \sin \left(\frac{A}{2}-0\right) \\
& =\sin ^{2} \frac{A}{2}-\sin ^{2} \theta . \\
\because 0 & \leqslant \sin ^{2} \theta \leqslant 1, \therefore \text { when } \sin ^{2} \theta=0 \text {, }
\end{aligned}
$$
$(\sin \alpha \sin \beta)$ maximum value $=\sin ^{2} \frac{A}{2}$;
when $\sin ^{2} \theta=1$, $(\sin \alpha \sin \beta)$ minimum value $=\sin ^{2} \frac{A}{2}-1$
$$
=-\cos ^{2} \frac{\mathrm{A}}{2} \text {. }
$$