4. In $\triangle A B C$, “ $\angle A\cos 2 B>\cos 2 C$ ” is a ( ) condition.
(A) neither sufficient nor necessary
(B) sufficient and necessary
(C) necessary but not sufficient
(D) sufficient but not necessary
4. B.

Notice,
$$
\begin{array}{l}
\angle A1-2 \sin ^{2} B>1-2 \sin ^{2} C \\
\Leftrightarrow \cos 2 A>\cos 2 B>\cos 2 C .
\end{array}
$$