2. (2004 College Entrance Examination - Zhejiang Paper) In $\triangle A B C$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and $\cos A=\frac{1}{3}$.
(1) Find the value of $\sin ^{2} \frac{B+C}{2}+\cos 2 A$;
(2) If $a=\sqrt{3}$, find the maximum value of $b c$.

保留了原文的换行和格式，如上所示。
2. (1) $\sin ^{2} \frac{B+C}{2}+\cos ^{2} A=\cos ^{2} \frac{A}{2}+\cos ^{2} A=\frac{1+\cos A}{2}+2 \cos ^{2} A-1=-\frac{1}{9}$;
(2) By the cosine rule $a^{2}=b^{2}+c^{2}-2 b c \cos A$ we get $3=b^{2}+c^{2}-2 b c \cdot \frac{1}{3} \geqslant 2 b c-\frac{2}{3} b c=\frac{4}{3} b c$ so $b c \leqslant \frac{9}{4}$, with equality if and only if $b=c$.