3.133. $2 \cos ^{2}\left(\frac{\alpha}{2}-\frac{3 \pi}{2}\right)+\sqrt{3} \cos \left(\frac{5}{2} \pi-\alpha\right)-1$.
## Solution.

$$
\begin{aligned}
& 2 \cos ^{2}\left(\frac{\alpha}{2}-\frac{3 \pi}{2}\right)+\sqrt{3} \cos \left(\frac{5}{2} \pi-\alpha\right)-1= \\
& =2 \cos ^{2}\left(\frac{3 \pi}{2}-\frac{\alpha}{2}\right)+\sqrt{3} \cos \left(\frac{5}{2} \pi-\alpha\right)-1= \\
& =2 \sin ^{2} \frac{\alpha}{2}-\sqrt{3} \sin \alpha-1=1-\cos \alpha+\sqrt{3} \sin \alpha-1=
\end{aligned}
$$

$=-\cos \alpha+\sqrt{3} \sin \alpha=-2\left(\frac{1}{2} \cos \alpha-\frac{\sqrt{3}}{2} \sin \alpha\right)=$

$=-2\left(\sin \frac{\pi}{6} \cos \alpha-\cos \frac{\pi}{6} \sin \alpha\right)=-2 \sin \left(\frac{\pi}{6}-\alpha\right)=2 \sin \left(\alpha-\frac{\pi}{6}\right)$

Answer: $2 \sin \left(\alpha-\frac{\pi}{6}\right)$.