Toss out means to release or to discard junk, rubbish, or trash that you no longer want. Usually you put it into the garbage, dump, or waste bin. In British English, they tend to use throw out in this context.
Ex: Why did you throw out all the letters on the table? The electric bill was there and I need to pay it soon. Now, how will I know how much to pay?
Ex: Xavier tossed out the tops of the tomatoes and onions because he couldn't use them in his cooking.
The phrase toss it aside means to pick something up and move it out of your way.
Ex: The head chef threw the chopped vegetables aside and started to prepare the meat and stew. He would add the vegetables later once the meat was fully cooked.
Ex: The tired student tossed the book aside and began to watch television. His brain was full of information and he needed to take a break from studying.
Cast aside usually refers to a person or ideas that someone has gotten rid of, rejected, or ignored because they weren't valuable to them.Cast originally comes from the Scandinavian word "kasta," meaning to throw.
Ex: The business leader cast aside the good advice of his employees and eventually the company failed because of his poor decisions.
Ex: After the woman became a famous celebrity, she cast aside all her old family and friends for the glitz and glamour of Hollywood.