$4.53 \frac{1+\cos (2 \alpha-2 \pi)+\cos (4 \alpha+2 \pi)-\cos (6 \alpha-\pi)}{\cos (2 \pi-2 \alpha)+2 \cos ^{2}(2 \alpha+\pi)-1}=2 \cos 2 \alpha$.

$$
\cos \left(4 \alpha-\frac{9 \pi}{2}\right)
$$
4.53 Using reduction formulas, as well as (4.17) and (4.21), we get

$$
\begin{aligned}
& \frac{1+\cos 2 \alpha+\cos 4 \alpha+\cos 6 \alpha}{\cos 2 \alpha+2 \cos ^{2} 2 \alpha-1}=\frac{2 \cos ^{2} \alpha+2 \cos 5 \alpha \cos \alpha}{\cos 2 \alpha+\cos 4 \alpha}= \\
& =\frac{\cos \alpha(\cos \alpha+\cos 5 \alpha)}{\cos 3 \alpha \cos \alpha}=\frac{2 \cos 3 \alpha \cos 2 \alpha}{\cos 3 \alpha}=2 \cos 2 \alpha
\end{aligned}
$$