<s>[INST] What is $\sin(nx)$ iteration in terms of $\sin A$ and $\cos A$? I want to use sum angle formulas, $\sin(A+B)=\sin A\cos B+\cos A\sin B$
to get for any angles, $\sin(nA)$ in terms of powers of $\sin A$ and $\cos A$.

I know there are other ways, but I want to use trigonometry and iteration on that. The goal is to carry out iteration in terms of $\sin(nA)$?

 [/INST] The angle addition formulas for sine and cosine are nicely captured by this matrix formula
$$
\left( \begin{array}{c} s_{n} \\ c_{n}\end{array} \right) =
\left( \begin{array}{r} c & s \\ -s & c\end{array} \right) 
\left( \begin{array}{c} s_{n-1} \\ c_{n-1}\end{array} \right)
$$
where $s_k = \sin(kx)$, $c_k = \cos(kx)$, $s= s_1$ and $c=c_1$.
Playing this recursion backwards, we arrive at the formula
$$
\left( \begin{array}{c} s_{n} \\ c_{n}\end{array} \right) =
\left( \begin{array}{r} c & s \\ -s & c\end{array} \right)^n
\left( \begin{array}{c} s_{0} \\ c_{0}\end{array} \right)
$$
Note that $s_0=0$ and $c_0=1$.
It's worth pointing out that that
$$
\left( \begin{array}{r} c & s \\ -s & c\end{array} \right) =
c \left( \begin{array}{r} 1 & 0 \\ 0 & 1\end{array} \right) +
s \left( \begin{array}{r} 0 & 1 \\ -1 & 0\end{array} \right)
$$
and that
$$
\left( \begin{array}{r} 0 & 1 \\ -1 & 0 \end{array}\right)^2 =  
\left( \begin{array}{r} -1 & 0 \\ 0 & -1\end{array} \right)
$$
</s>