(1)

\begin{align*} T_{n}(x) & =\frac{1}{2}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}+\left(x-i\sqrt{1-x^{2}}\right)^{n}\right)\\ & =\frac{1}{2}\sum_{k=0}^{n}\left(C(n,k)\left(i\sqrt{1-x^{2}}\right)^{k}x^{n-k}+C(n,k)\left(-i\sqrt{1-x^{2}}\right)^{k}x^{n-k}\right)\\ & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k)\left(i\sqrt{1-x^{2}}\right)^{2k}x^{n-2k}\right)\\ & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k)\left(-1\right)^{k}\left(1-x^{2}\right)^{k}x^{n-2k}\right) \end{align*}

(2)

\begin{align*} U_{n-1}(x) & =\frac{1}{2i\sqrt{1-x^{2}}}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}-\left(x-i\sqrt{1-x^{2}}\right)^{n}\right)\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\sum_{k=0}^{n}\left(C(n,k)\left(i\sqrt{1-x^{2}}\right)^{k}x^{n-k}-C(n,k)\left(-i\sqrt{1-x^{2}}\right)^{k}x^{n-k}\right)\\ & =\frac{1}{i\sqrt{1-x^{2}}}\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k+1)\left(i\sqrt{1-x^{2}}\right)^{2k+1}x^{n-2k-1}\right)\\ & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k+1)(-1)^{k}\left(1-x^{2}\right)^{k}x^{n-2k-1}\right) \end{align*}