$4.46 \frac{\operatorname{ctg}^{2}(2 \alpha-\pi)}{1+\operatorname{tg}^{2}\left(\frac{3 \pi}{2}-2 \alpha\right)}-3 \cos ^{2}\left(\frac{5 \pi}{2}-2 \alpha\right)=$

$=4 \sin \left(\frac{\pi}{6}-2 \alpha\right) \sin \left(\frac{\pi}{6}+2 \alpha\right)$.
4.46 After transformations, the left side will take the following form:

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

## Further, we have

$$
\begin{aligned}
& A=4\left(\sin \frac{\pi}{6} \cos 2 \alpha-\cos \frac{\pi}{6} \sin 2 \alpha\right)\left(\sin \frac{\pi}{6} \cos 2 \alpha+\cos \frac{\pi}{6} \sin 2 \alpha\right)= \\
& =4 \sin \left(\frac{\pi}{6}-2 \alpha\right) \sin \left(\frac{\pi}{6}+2 \alpha\right)
\end{aligned}
$$