(1)

\begin{align*} U_{2n-1}(x) & =\frac{\sin(2n\cos^{\bullet}x)}{\sin\cos^{\bullet}x}\\ & =\frac{2\sin(n\cos^{\bullet}x)\cos(n\cos^{\bullet}x)}{\sin\cos^{\bullet}x}\\ & =2U_{n-1}(x)T_{n}(x) \end{align*}

(2)

\begin{align*} U_{2n}(x) & =2^{2n}\prod_{k=1}^{2n}\left(x-\cos\frac{k\pi}{2n+1}\right)\\ & =2^{2n}\prod_{k=1}^{n}\left(x-\cos\frac{2k\pi}{2n+1}\right)\prod_{k=1}^{n}\left(x-\cos\frac{(2k-1)\pi}{2n+1}\right)\\ & =2^{2n}\prod_{k=1}^{n}\left(x-\cos\frac{2k\pi}{2n+1}\right)\prod_{k=1}^{n}\left(x+\cos\left(\pi-\frac{(2k-1)\pi}{2n+1}\right)\right)\\ & =2^{2n}\prod_{k=1}^{n}\left(x-\cos\frac{2k\pi}{2n+1}\right)\prod_{k=1}^{n}\left(x+\cos\left(\frac{2(n-k+1)\pi}{2n+1}\right)\right)\\ & =2^{n}\prod_{k=1}^{n}\left(x-\cos\frac{2k\pi}{2n+1}\right)2^{n}\prod_{k=1}^{n}\left(x+\cos\frac{2k\pi}{2n+1}\right)\\ & =(-1)^{n}2^{n}\prod_{k=1}^{n}\left(x-\cos\frac{k\pi}{n+\frac{1}{2}}\right)2^{n}\prod_{k=1}^{n}\left(-x-\cos\frac{k\pi}{n+\frac{1}{2}}\right)\\ & =(-1)^{n}W_{n}(x)W_{n}(-x) \end{align*}