(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) $\mathrm{C}$ Hint: $|x+y|+|x-y|=2|x|$ or $2|y|$, so when $|x|<1$ and $|y|<1$, $|x+y|+|x-y|<2$; also
$$
\begin{array}{l}
|x|=\frac{1}{2}|x+y+x-y| \leqslant \frac{1}{2}(|x+y|+|x-y|), \\
|y|=\frac{1}{2}|x+y-(x-y)| \leqslant \frac{1}{2}(|x+y|+|x-y|),
\end{array}
$$

so when $|x+y|+|x-y|<2$, $|x|<1$ and $|y|<1$. Therefore, it is a sufficient and necessary condition.