(1)

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

(2)

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