10. Proof: $\left\{\frac{\theta}{\pi}\right\}=0$ is obviously true. When $\left\{\frac{\theta}{\pi}\right\} \neq 0$, from (41) and (42) we get
$$\begin{array}{l} \sin \alpha+\sin 3 \alpha+\cdots+\sin (2 n-1) \alpha \\ \quad=\frac{\sin n \alpha}{\sin \alpha} \sin n \alpha \\ \cos \alpha+\cos 3 \alpha+\cdots+\cos (2 n-1) \alpha \\ \quad=\frac{\sin n \alpha}{\sin \alpha} \cos n \alpha \end{array}$$

Dividing the two equations yields
$$\operatorname{tg} n \alpha=\frac{\sin \alpha+\sin 3 \alpha+\cdots+\sin (2 n-1) \alpha}{\cos \alpha+\cos 3 \alpha+\cdots+\cos (2 n-1) \alpha}$$