4.22

$$
\frac{\cos ^{2}\left(\frac{5 \pi}{4}-2 \alpha\right)-\sin ^{2}\left(\frac{5 \pi}{4}-2 \alpha\right)}{\left(\cos \frac{\alpha}{2}+\sin \frac{\alpha}{2}\right)\left(\cos \left(2 \pi-\frac{\alpha}{2}\right)+\cos \left(\frac{\pi}{2}+\frac{\alpha}{2}\right)\right) \sin \alpha}
$$
4.22 Applying formula (4.14) to the numerator and then the reduction formula, we get $\cos \left(\frac{5 \pi}{2}-4 \alpha\right)=\sin 4 \alpha$. Transform the product of the first two factors in the denominator:

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

Finally, we find

$\frac{\sin 4 \alpha}{\cos \alpha \sin \alpha}=\frac{2 \sin 2 \alpha \cos 2 \alpha}{0.5 \sin 2 \alpha}=4 \cos 2 \alpha$.