$4.19 \frac{\sin ^{2}\left(\frac{\pi}{2}+\alpha\right)-\cos ^{2}\left(\alpha-\frac{\pi}{2}\right)}{\tan^{2}\left(\frac{\pi}{2}+\alpha\right)-\cot^{2}\left(\alpha-\frac{\pi}{2}\right)}$.
### 4.19 We have

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

We used the reduction formulas, as well as (4.14), (4.3), (4.4), and (4.13).