4. In $\triangle A B C$, “ $A\cos 2 B>\cos 2 C$ ” is
A. Neither a sufficient nor a necessary condition
B. A sufficient and necessary condition
C. A necessary but not a sufficient condition
D. A sufficient but not a necessary condition
$$
\cos 2 A>\cos 2 B>\cos 2 C \Leftrightarrow \sin ^{2} A<\sin ^{2} B<\sin ^{2} C \Leftrightarrow \sin A<\sin B<\sin C
$$
$\Leftrightarrow a<b<c \Leftrightarrow A<B<C$, so the answer is $B$.