1. Let $A B C D$ be a cyclic quadrilateral, and now four relations are given:
(1) $\sin A=\sin C$;
(2) $\sin A+\sin C=0$;
(3) $\cos B+\cos D=0$;
(4) $\cos B=\cos D$.

The number of relations that always hold is
(A) one;
(B) two;
(C) three;
(D) four.
1. (B)

Since $A B C D$ is a cyclic quadrilateral, $C=180^{\circ}-A$, and $A$ and $C$ cannot be $0^{\circ}$ or $180^{\circ}$, therefore equation (1) always holds; equation (2) never holds.

Similarly, since $D=180^{\circ}-B$, equation (3) always holds; equation (4) only holds when $B=D=90^{\circ}$. Therefore, among the four given equations, only (1) and (3) always hold.