$4.55 \cos ^{6} \alpha+\sin ^{6} \alpha=\frac{5+3 \cos 4 \alpha}{8}$. 

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$4.55 \cos ^{6} \alpha+\sin ^{6} \alpha=\frac{5+3 \cos 4 \alpha}{8}$.
4.55 Using the formula for the sum of cubes, we get $\cos ^{6} \alpha+\sin ^{6} \alpha=\left(\cos ^{2} \alpha+\sin ^{2} \alpha\right)\left(\cos ^{4} \alpha-\cos ^{2} \alpha \sin ^{2} \alpha+\sin ^{4} \alpha\right)=$

$$
\begin{aligned}
& =\left(\frac{1+\cos 2 \alpha}{2}\right)^{2}-\frac{1}{4} \sin ^{2} 2 \alpha+\left(\frac{1-\cos 2 \alpha}{2}\right)^{2}= \\
& =\frac{1}{4}\left(1+2 \cos 2 \alpha+\cos ^{2} 2 \alpha-\sin ^{2} 2 \alpha+1-2 \cos 2 \alpha+\cos ^{2} 2 \alpha\right)= \\
& =\frac{1}{4}\left(1+3 \cos ^{2} 2 \alpha\right)=\frac{1}{8}(5+3 \cos 4 \alpha)
\end{aligned}
$$

Here, formulas (4.16), (4.17), and (4.13) were also used.