4.54

$$
\operatorname{ctg}\left(\frac{5 \pi}{4}+2 \alpha\right)\left(1-\cos \left(\frac{5 \pi}{2}+4 \alpha\right)\right)=\operatorname{tg} 4 \alpha
$$
4.54 Transform the left side:

$\frac{\sin 4 \alpha}{\operatorname{ctg}\left(\frac{\pi}{4}+2 \alpha\right) \cdot 2 \sin ^{2}\left(\frac{5 \pi}{4}+2 \alpha\right)}=$

$=\frac{\sin 4 \alpha}{2 \operatorname{ctg}\left(\frac{\pi}{4}+2 \alpha\right) \cdot \sin ^{2}\left(\frac{\pi}{4}+2 \alpha\right)}=$

$=\frac{\sin 4 \alpha}{\sin \left(\frac{\pi}{2}+4 \alpha\right)}=\frac{\sin 4 \alpha}{\cos 4 \alpha}=\operatorname{tg} 4 \alpha$.