What you will do, when your boss gave you a 'waste of time' task with a countdown? How will you respond?

Do it immediately.
You can think it's a waste of time but there is a countdown so, it won't be such a waste of time matter.
Your boss knows more than employees and maybe you are doing something important for him probably he won't tell you.
But he plays his cards.
And another reason is because he pays your salary so, you have to do it.

Boss is a boss and he knows why you have been assigned a particular task.
He may be having some good reasons of which you may not be aware.
So the conclusion is "go ahead with the task."

I think it will be excellent chance to insure the convenience my boss as will as reflect his trust in me

Do it because may be he has given to keep you busy while you are having nothing to do in your idle working time.

I will try to convince the boss.
If he does not agree I will start doing the task immediately as the "boss is always right"