$4.25 \sin 4 \alpha-2 \cos ^{2} 2 \alpha+1$.

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$4.25 \sin 4 \alpha-2 \cos ^{2} 2 \alpha+1$.
4.25 From formula (4.17), it follows that $2 \cos ^{2} 2 \alpha-1=\cos 4 \alpha$. Then the given expression will take the form $A=\sin 4 \alpha-\cos 4 \alpha$ and, therefore,

$A=\sin 4 \alpha-\sin \left(90^{\circ}-4 \alpha\right)=2 \cos 45^{\circ} \sin \left(4 \alpha-45^{\circ}\right)=$ $=\sqrt{2} \sin \left(4 \alpha-45^{\circ}\right)$.