General form for $\sin(kx)$ in terms of $\sin(x)$ and $\cos(x)$ Identities for $\sin(2x)$ and $\sin(3x)$, as well as their cosine counterparts are very common, and can be used to synthesize identities for $\sin(4x)$ and above. Given some integer $k$, is there an equation to find $\sin(kx)$ in terms of $\sin(x)$ and $\cos(x)$? For example, if I wanted to find the identity for $\sin(1000x)$, what would it be? Identies used: $$\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)\\\\\cos(a+b)=\cos(a)\cos(b)+\sin(a)\sin(b)$$

Use the De Moivre formula to calculate $\sin(nx)$ and $\cos(nx)$ for large $n$