Solving for $\sin(n \cdot\arcsin(x))$ when n takes on any integer value. 

Solving for $\sin(n \cdot\arcsin(x))$ when n takes on any integer value.
How would I go about tackling this question for large numbers of n? I was able to compute it by hand for values of $n$ up to $4$ but there has to be an easier way to solve it for numbers like $50$ etc...



If $U_n$ denotes de $n-$th Chebyshev polynomial of the second kind, we have :
$$\forall\theta\in\mathbb{R},\quad\sin((n+1)\theta)=\sin(\theta)\,U_{n}(\cos(\theta))$$
So, for all $x\in[-1,1]$ and $n\in\mathbb{N}^\star$ :
$$\sin(n\arcsin(x))=x\,U_{n-1}\left(\sqrt{1-x^2}\right)$$
It remains to explain how to compute effectively $U_n(t)$ for every pair $(n,t)$.
A simple way is to use the recurrence relation :
$$U_0(t)=1\qquad U_1(t)=2t\qquad \forall n\ge2,\,U_n(t)=2t\,U_{n-1}(t)-U_{n-2}(t)$$
