4. In $\triangle A B C$, $A>B$ is ( ) of $\cos 2 B>\cos 2 C$.
(A) A sufficient but not necessary condition.
(B) A necessary but not sufficient condition.
(C) A sufficient and necessary condition.
(D) Neither a sufficient nor a necessary condition.
$\begin{array}{l}\text { 4. (C). } \\ \triangle A B C \text {, in which } A>B \Leftrightarrow a>b \Leftrightarrow 2 R \sin A> \\ 2 R \sin B \Leftrightarrow \sin A>\sin B>0 \Leftrightarrow \sin ^{2} A>\sin ^{2} B \Leftrightarrow \\ -2 \sin ^{2} A1- \\ 2 \sin ^{2} A \Leftrightarrow \cos ^{2} B>\cos ^{2} A .\end{array}$