Example 8. Find the sum: $\cos \alpha+\cos 3 \alpha+\cos 5 \alpha+\cdots$ $+\cos (2 n-1) \alpha$.
Solve the original expression $\begin{array}{l}=\frac{1}{2 \sin \alpha}[2 \sin \alpha \cos \alpha+2 \sin \alpha \cos 3 \alpha \\ +2 \sin \alpha \cos 5 \alpha+\cdots+2 \sin \alpha \cos (2 n-1) \alpha] \\ =\frac{1}{2 \sin \alpha}[\sin 2 \alpha \div(\sin 4 \alpha-\sin 2 \alpha) \\ +(\sin 6 \alpha-\sin 4 \alpha)+\cdots+\sin 2 n \alpha \\ -\sin 2(n-1) \alpha]=\frac{\sin 2 n \alpha}{2 \sin \alpha} .\end{array}$