how does the rising and sinking of air (high and low pressure) in the atmosphere cause ....?

Question

a. clear and rainy days
b. deserts near 30 degrees N and 30 degrees S
c. rain shadows
d. the monsoon of southern Asian and the southwestern United States

Answer 1

The equator is hot and air rises (low pressure). The poles are cold and the air sinks (high pressure). If it wasn't for the earth rotation and the resulting Coriolis effect, it would be like that. But the air masses trying to even out lows and high create again a high pressure belt around latitude 30N (or south) and a low pressure belt around latitude 60N (or south).
Air that sinks in a high pressure gets warmer (adiabatic effect, like when you pump the tyres of your bicycle). Moisture in that air dissipate because warm air can contain more humidity. The sky is clear.
Air that rises cools down for the same adiabatic effect and the moisture content of the air condenses in clouds then rain or snow.
The lows around the equator are simple convections due to the extreme heat and the supply of humidity from the oceans. Lows around the temperate zones of the latitudes 60 are frontal lows. Those fronts are the separation between polar dry and cold air, and the temperate and humid air from the south (north, in the southern hemisphere).
This is in a nutshell. The complete answer would be in the meteorology course I teach pilots and that lasts four hours.