Polynomials defined from sines of multiples of an inverse cosine, and a three-term recurrence

For \(-1<x<1\) and each integer \(n\geq0\), the function \(U_n\) is defined by \(U_n(x)=\frac{\sin\left((n+1)\arccos x\right)}{\sin(\arccos x)}\).

1. Find \(U_0(x)\).[1]
2. Show that \(U_1(x)=2x\).[2]
3. 1. Simplify \(\sin(A+B)+\sin(A-B)\).
   2. Hence show that \(U_{n+1}(x)+U_{n-1}(x)=2xU_n(x)\) for \(n\geq1\).[5]
4. Express \(U_3(x)\) as a polynomial in \(x\).[2]