Pythagorean triplet mutiple angles If both cos and sin are rational, then, is sin(nx) rational?
From just checking for obvious values other than, 0,1
For tan(x)=3/4,  for n=2, we get another pythagorean triplet of 24,7,25, for n=3, 336,527,625, for n=4, 354144,164833,390625..
If this trend continuous, is there a De moivre proof? I would like to stay away from number theory if possible.

Since $\mathbb Q[i]$ (the set of complex numbers with rational real and imaginary parts) is closed under multiplication, it follows directly from de Moivre's formula that if $\cos x$ and $\sin x$ are both rational, then so are $\cos(nx)$ and $\sin(nx)$ for all integer $n$.
