3.144. $3+4 \cos 4 \alpha+\cos 8 \alpha$.

3.144. $3+4 \cos 4 \alpha+\cos 8 \alpha$.
## Solution.

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

Answer: $8 \cos ^{4} 2 \alpha$.