Solution:

Since $\sin t + i\cos t = i(\cos t - i \sin t)$ and $\sin nt + i\cos nt = i(\cos nt - i\sin nt),$ de Moivre's theorem (applied to angle $-t$) gives $$(\sin t + i \cos t)^n = i^n(\cos t - i\sin t)^n = i^n(\cos nt - i \sin nt).$$

So the equation holds for all real $t$ exactly when $i^n = i,$ that is, when $n \equiv 1 \pmod 4.$ The values $n = 1, 5, 9, \ldots, 997$ give exactly $250$ positive integers up to $1000.$