The answer is almost like "you have a nose and a mouth for the same reason" - the torso is sort of like an extended nose (and upper lip), and the nose and mouth serve completely different purposes. The elephant's mouth is mainly used for chewing and swallowing. It contains teeth and tusks, if any (these are the incisors, which are long and protrude from the mouth on either side of the torso), as well as a tongue. The trunk, on the other hand, is used to lift, hold, smell, dust itself, pat baby elephants, make sounds, shower, greet other elephants (wrap and caress), and of course, while the nostrils are working Breathe all the way inside.
When eating and drinking, both the trunk and the mouth are used, but for different purposes. The trunk selects and prepares a bite of food and then passes it to the mouth for chewing and swallowing. Drinking too. Elephants don't drink water through their trunks, just like you don't drink water through your trunks. Instead, the trunk absorbs water (which can hold about ten liters) and blows the water into the mouth to swallow. So elephants have trunks and mouths because they need both.
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