Did Trigonometry just fail?

$a+\dfrac{1}{a} = 2\cos{6^\circ} \quad \Rightarrow a^2 - 2 \cos{6^\circ}a + 1 = 0$ $\Rightarrow a = \dfrac {2\cos{6^\circ} \pm \sqrt{4\cos^2{6^\circ}-4}}{2} = \cos{6^\circ} \pm i \sin{6^\circ} = e^{\pm6^\circ i}$

$\begin{aligned} \Rightarrow a^{1000}+\dfrac{1}{a^{1000}} + 1 & = e^{6000^\circ i} +e^{-6000^\circ i} + 1 = 2\cos {6000^\circ } + 1 \\ & = 2\cos {240^\circ } + 1 = 2 \left( - \frac {1}{2} \right) + 1 \\ & = -1 + 1 = \boxed{0} \end{aligned}$

This!