Example 1. Find $P_{n}=\prod_{k=1}^{n}\left(2 \cos 2^{k-1} \theta-1\right)$.
Solve $\begin{aligned} p_{k} & =2 \cos 2^{k-1} \theta-1 \\ & =\frac{\left(2 \cos 2^{k-1} \theta-1\right)\left(2 \cos 2^{\mathbf{k}-1} \theta+1\right)}{2 \cos 2^{k-1} \theta+1} \\ & =\frac{4 \cos ^{2} 2^{\mathbf{k}-1} \theta-1}{2 \cos 2^{\mathbf{k}-1} \theta+1} \\ & =\frac{2\left(\cos { }^{K} \theta+1\right)-1}{2 \cos 2^{K-1} \theta+1} \\ & =\frac{2 \cos 2^{K} \theta+1}{2 \cos 2^{K-1} \theta+1} \\ \therefore P_{n} & =(2 \cos \theta-1)(2 \cos 2 \theta-1) \\ & \cdot(2 \cos 4 \theta-1) \cdots\left(2 \cos 2^{\mathbf{n}-1} \theta-1\right) \\ & =\frac{2 \cos 2 \theta+1}{2 \cos \theta+1} \cdot \frac{2 \cos 4 \theta+1}{2 \cos 2 \theta+1}\end{aligned}$