In anticipation of Rosh Ha-Shana we take haircuts and dress as we do for Yom Tov to demonstrate that we are confident that G-d will be kind to us and judge us favorably on the Day of Judgement. Likewise, just after the close of the Yom Ha-Kippurim we celebrate by having a festive feast, to demonstrate again that we are confident that the judgement was a favorable one.
How can we be so confident? Every year tragedies do occur. Some young people die prematurely; others become impoverished. There is a lot of suffering in the world that would seem to contradict such confidence.
The Chazon Ish explains in his essay on Emunah and Bitachon, that when we ask a sick person to have bitachon, it does not mean that he should be convinced that he will recover. That would be ridiculous - one can not be sure that he will not die. Bitachon simply means to live by emunah, and emunah means believing that G-d has complete control over everything in the world. If G-d wants me to live and be healthy and happy, then there is nothing anyone can do to negate that. If for some reason, G-d wants me to suffer, then as that is His will, we should accept it with joy, with the knowledge that anything G-d does is for the good.
When someone harms another, one should not think that were it not for that individual the first person would not have suffered. We believe that bechira is always limited. No one has the ability to harm another person unless there was a gezeirah from heaven decreeing that the victim should be harmed. "No one will injure his finger on this earth unless it was so ordained from above" (Chulin 7b}. Once such a gezeirah is decreed, G-d gives everyone the ability to use his or her bechirah even to the extent of harming another individual.
We are all expected to lead our lives in accordance with these principles of emunah, and living by the principles of emunah is what it means to have bitachon. We are not confident that our judgement on Rosh Ha-Shana and Yom Ha-Kippurim will be in our favor. Rather, our confidence consists of our belief that G-d is all-powerful, that His will shall prevail, and that whatever He does is always letov, even in the event that it is detrimental to us.