Example 2 Let $n=1999$. Find the value of the following expression:
$$
\frac{1}{2^{n}}\left(1-3 \mathrm{C}_{n}^{2}+3^{2} \mathrm{C}_{n}^{4}-3^{3} \mathrm{C}_{n}^{6}+\cdots-3^{999} \mathrm{C}_{n}^{1998}\right) \text {. }
$$
$\begin{array}{l}\text { Solution: Original expression }=\frac{1}{2^{n+1}}\left[(1+\sqrt{3} i)^{n}+(1-\sqrt{3} i)^{n}\right] \\ =\frac{1}{2}\left[\left(\frac{1}{2}+\frac{\sqrt{3}}{2} i\right)^{n}+\left(\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)^{n}\right) \\ =\frac{1}{2}\left[\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)^{n}+\left(\cos \frac{\pi}{3}-i \sin \frac{\pi}{3}\right)^{n}\right] \\ =\cos \frac{n \pi}{3}=\cos \frac{1999 \pi}{3}=\frac{1}{2} .\end{array}$