（湘潭大学，2006 年）计算 $n(n \geqslant 2)$ 阶行列式：

$$
D=\left|\begin{array}{cccc}
\sin \left(2 \alpha_1\right) & \sin \left(\alpha_1+\alpha_2\right) & \cdots & \sin \left(\alpha_1+\alpha_n\right) \\
\sin \left(\alpha_2+\alpha_1\right) & \sin \left(2 \alpha_2\right) & \cdots & \sin \left(\alpha_2+\alpha_n\right) \\
\vdots & \vdots & & \vdots \\
\sin \left(\alpha_n+\alpha_1\right) & \sin \left(\alpha_n+\alpha_2\right) & \cdots & \sin \left(2 \alpha_n\right)
\end{array}\right| .
$$