When james sprayed liquid deodorant under his arm, the spot felt cold. which sentence best explains why?

There is no correct sentence on the list you provided.  So I'll have to
explain it in my own words: 

Inside the can, along with the deodorant, is a 'propellent' ... some substance
that makes it spray out when there's a tiny hole in the can. 

The propellent is under pressure in the can, to make it really WANT to
come out when there's a way to get out.

When the hole opens up, the propellent whooshes out, and carries some
deodorant out with it.  Of course, when this is happening, the pressure on
the propellent is dropping ... from the high pressure inside the can, to the
normal atmospheric pressure outside the can.

What happens when the pressure on a gas decreases ?   The temperature
of the gas drops.  So, by the time the stuff from inside the can hits your skin,
it has cooled suddenly, because its pressure has dropped suddenly.