$4.62 \cos ^{4} 2 \alpha-6 \cos ^{2} 2 \alpha \sin ^{2} 2 \alpha+\sin ^{4} 2 \alpha$.
### 4.62 Finding

\[
\begin{aligned}
& \cos ^{4} 2 \alpha-6 \cos ^{2} 2 \alpha \sin ^{2} 2 \alpha+\sin ^{4} 2 \alpha= \\
& =\left(\cos ^{2} 2 \alpha-\sin ^{2} 2 \alpha\right)^{2}-\sin ^{2} 4 \alpha=\cos ^{2} 4 \alpha-\sin ^{2} 4 \alpha=\cos 8 \alpha
\end{aligned}
\]