Do not give in! You are not asking your children to do too much by having chores in the morning and sharing working in in the garden in the evening.
You are the adult and his tantrums are having the exact effect he wants. He's wearing you down. He's making you question yourself. Do not give in!
His behavior is not going to change as long as he thinks there's a chance he'll get his way. He'll keep "ratching it up!" Tell him, what you expect and you won't give in. Tell him you really wanted to take him to the museum, but because of his behavior you can't go. Remind him the event has to be a pleasant experience for you too, otherwise, you won't do that event again. You have to willing to stick with whatever you say. If he embarrasses you in the grocery store, tell him he won't come to the grocery store with you again and work it out! He has to know you mean business and YOU, not him, are in charge.
What does he value? TV? Time alone? Juice? Games? Whatever it is, I'd take it away until he improves. My son is 4. He loves to be read to. He had so many "red" days at daycare, I took away his books and he had to earn them back one by one. Every time he had a green day he got to pick one book to return to his room. Worked like a charm, after a few tantrums.
Taking away his juice and milk and giving him water for the rest of the day use to work. Now, he likes water and tells me it "makes him strong." So, I had to move on an find something else he valued.
Another suggestion, is let the tantrum go. STAY CALM! Once it's over, he'll want something. Tell him, "After a tantrum like that you don't get what you want." He'll throw another tantrum and wear himself out. He'll ask for something. Tell him, "No, after a tantrum like that, you do't get what you want." This is hard, hard, hard to do and you can't really be under a time constraint for this approach to work in the short run.
This is not easy! It will take time, in the middle of a 30-40 minute tantrum, it will seem like FOREVER.
Look for small improvements, like having a 35 minute tantrum instead of a 40 minute tantrum.
Be strong, my friend. You will thank yourself a year from now. Yes, I said a year! Sorry this is so long. I'm very passionate about behavior.