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

$$
\begin{aligned}
& \sin \left(\frac{5}{2} \pi-2 \alpha\right)+2 \sin ^{2}\left(2 \alpha-\frac{3}{2} \pi\right)-1=\sin \left(\frac{5}{2} \pi-2 \alpha\right)+2 \sin ^{2}\left(\frac{3}{2} \pi-2 \alpha\right)-1= \\
& =\cos 2 \alpha+2 \cos ^{2} 2 \alpha-1=\cos 2 \alpha+\cos 4 \alpha=2 \cos 3 \alpha \cos \alpha= \\
& =2 \cos \alpha \cos 3 \alpha
\end{aligned}
$$

Answer: $2 \cos \alpha \cos 3 \alpha$.