Should I follow my dad around to see if he’s cheating on my mom?

What makes you think that he's cheating? Or that if he was, and you could prove it, that your mum would leave him for good? It's very hard for people to get out of abusive relationships, especially when there's financial and familial ties (children). They rationalize the abuse by thinking that the behaviour is normal, they deserve it and have low self-worth, and/or can't leave because of money and family.
What would happen to you if he caught you spying? It could put you in a dangerous spot. It could also back fire by him blaming your mum for spying. There is a risk of him becoming aggressive. Of course I'm not saying it would be your fault for provoking him, because the behaviour is wrong and has no justification, but you and your mum's safety is top priority. I'd advise you against spying, that's my opinion. x
Getting people out of abusive relationships is tricky, but the key is having a support network and not forcing the person to do anything- that includes leave the abusive partner. I know that seems the opposite of what you should do, but they need to decide to do that on their own and you just need to support her in the process. Think of it this way, if you try to force her to leave him, it's just one more thing, one more person trying to control her. If you have siblings, talk to them about it too, trusted family members etc, and just get her doing things away from him. There are lots of little things you and other supporters can do to help build her self-esteem and independence up. Take her out for coffee and don't talk about him, talk about anything else. Go for a drive without him, the beach if you live close. Take over chores she would normally do and let her know she deserves a break. Encourage her to go out with her friends. Just let her know she is loved and deserves love. Once you think she's in a good state of mind, that's when you can bring up and express concern for her and that you oppose your dad's aggressive behaviour and that it's not okay. Again though, remember never to force her. It's about giving her the tools (self-esteem, independence, and information) she needs to make the right choice. I hope this helps.

It doesn't seem like it matters whether or not he is cheating. This is your mom's problem, and the only thing you should concern yourself with is to support her where you can, and be a good son. Throwing proof of things she already knows won't help her. It most likely will only strain your relationship with her, isolating her from someone she should be able to trust and rely on.

You could, im not saying you should, Film your dad hitting your mom. Without him seeing it. Film all the bad things he does and keep it as evidence, then go to a police station with it and explain the whole thing.You could do that, or leave it to your mom since she's the one with the problems

I would not really want to see him fuck..its his life man, im sorry..