Trade in value/ tax credit?

Can someone please help me understand how this works? I own my car and want to trade in when I get a new lease. I want to take the entire amount ,$5k, and get a check for it. Put nothing down. Someone told me I can use my trade in value to offset the tax I pay. Is that true if I’m not putting any $ down? I dnt know how that works. I’m in New Jersey if that make a difference

In Pennsylvania, when I traded my car in, I did NOT get a standard tax credit paid back to me for my lease like you would normally get in a purchasing situation. I don’t know how NJ works but that is my experience.