3.114. $\sin 4 \alpha-2 \cos ^{2} 2 \alpha+1$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.

3.114. $\sin 4 \alpha-2 \cos ^{2} 2 \alpha+1$.
Solution.

$$
\begin{aligned}
& \sin 4 \alpha-\cos 4 \alpha=\sin 4 \alpha-\sin \left(90^{\circ}-4 \alpha\right)=2 \cos 45^{\circ} \sin \left(4 \alpha-45^{\circ}\right)= \\
& =2 \cdot \frac{\sqrt{2}}{2} \cdot \sin \left(4 \alpha-45^{\circ}\right)=\sqrt{2} \sin \left(4 \alpha-45^{\circ}\right)
\end{aligned}
$$

Answer: $\sqrt{2} \sin \left(4 \alpha-45^{\circ}\right)$.