I am not sure what you are asking.

But sometimes I get expressions which look like,

.

And in general there is no way to simplify it, you just leave it like that. Part of the difficulty in the word "simplify" is that it is not well-defined, and hence we cannot come up with a theory which tells us when we can "simplify" and when we cannot. Perhaps you can create a definition.

And, I also do not understand how you got trigonometric functions in your answer. I never remember learning it that.