1. Given $x \in \mathbf{R}, y \in \mathbf{R}$. Then “ $|x|<1$, and $|y|<1$ ” is “ $|x+y|+|x-y|<2$ ” ( ).
(A) a sufficient condition but not a necessary condition
(B) a necessary condition but not a sufficient condition
(C) a sufficient and necessary condition
(D) neither a sufficient condition nor a necessary condition
1. C.

Notice that,
$$
|x+y|+|x-y|=2 \max \{|x|,|y|\} \text{.}
$$

Therefore, it is a sufficient and necessary condition.