Why is it sunny when the air pressure is low before the cold front crosses the border?

1. The low air pressure mentioned here is relative to the fact that the place is controlled by a cold air mass and the air pressure rises after the cold front passes through. In other words, the air pressure is the result of the comparison between the two situations.

It often rains near the low-pressure center and trough, not all low-pressure parts have rain.

2. When a single air mass is controlled, the atmospheric properties are more uniform and stable, so it is sunny. Only when the air mass moves upward will there be clouds and rain.