Your nose senses something that doesn't belong and wants to rid the body of it.
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When you shiver after sneezing, it either means you are sick or your sneezing sent a cold chill down your spine when all the warmth came out. When a cold chill goes down your spine, it just means you got really cold all of a sudden and your body is reacting naturally. Hear is the best way to beat a cold or flu. First eat lots of chicken noodle soup. ( Unless you are allergic to it.) If you are then just make sure you take your medicine +when you should. Try to stay in bed and blow your nose often.
Possibly while coughing something got pushed up into the nasal cavities causing you to sneeze.
because the pressure of the sneeze is pretty much everywhere on your body, which pushes out the fart