(1)

\begin{align*} \int_{-1}^{1}V_{m}(x)V_{n}(x)\sqrt{\frac{1+x}{1-x}}dx & =-\int_{\pi}^{0}V_{m}(\cos t)V_{n}(\cos t)\sqrt{\frac{1+\cos t}{1-\cos t}}\sin tdt\qquad,\qquad x=\cos t\\ & =\int_{0}^{\pi}\frac{\cos\left(\left(m+\frac{1}{2}\right)t\right)}{\cos\left(\frac{1}{2}t\right)}\frac{\cos\left(\left(n+\frac{1}{2}\right)t\right)}{\cos\left(\frac{1}{2}t\right)}\sqrt{\frac{1+\cos t}{1-\cos t}}\sin tdt\\ & =\int_{0}^{\pi}\cos\left(\left(m+\frac{1}{2}\right)t\right)\cos\left(\left(n+\frac{1}{2}\right)t\right)\sqrt{\frac{1+\cos t}{1-\cos t}}\frac{2\sin t}{1+\cos t}dt\\ & =2\int_{0}^{\pi}\cos\left(\left(m+\frac{1}{2}\right)t\right)\cos\left(\left(n+\frac{1}{2}\right)t\right)dt\\ & =\int_{0}^{\pi}\left\{ \cos\left(\left(m+n+1\right)t\right)+\cos\left(\left(m-n\right)t\right)\right\} dt\\ & =\pi\delta_{mn} \end{align*}

(2)

\begin{align*} \int_{-1}^{1}W_{m}(x)W_{n}(x)\sqrt{\frac{1-x}{1+x}}dx & =-\int_{1}^{-1}W_{m}(-x)W_{n}(-x)\sqrt{\frac{1+x}{1-x}}dx\\ & =(-1)^{m+n}\int_{-1}^{1}(-1)^{m}W_{m}(-x)(-1)^{n}W_{n}(-x)\sqrt{\frac{1+x}{1-x}}dx\\ & =(-1)^{m+n}\int_{-1}^{1}V_{m}(x)V_{n}(x)\sqrt{\frac{1+x}{1-x}}dx\\ & =\pi\delta_{mn} \end{align*}