are core workouts necessary?

I'm 5'11'' 170 lbs., I have a beer belly, I'm trying to get rid of that and gain some upper body strength. My personal trainer has me doing a full body workout and I have to do some core body workouts. Do I have to do this? Or why can't I just lift weights and run cardio? Some of the core workouts I don't like are her leg exercies since she says she doesn't like those leg machines since it's bad for your knees which Im' sure it is, I've experienced that, but I don't like having to workout my legs with the exercies she showed me. I feel weird and uncomfortable doing those.
 
Core work is extremely important. All movement starts from the core, so it needs to be strengthened. What are core leg exercies?
 
What are core leg exercies?

Kind of like you're about to sit on a toilet, you try to touch the edge of a chair like that with your rear outwards and keep your back straight. I don't know if 'leg exercies' is the correct term I mentioned but that was one of the exercises. Another one was where I step up on something kind of like a stair, and stand up on one leg and hold it.
 
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