Why half coversed or coversed trigonometric functions are being deprecated? As you can see here there are some names for some trigonometric functions that I can't find in any text or math related papers today.
In my opinion this kind of approach will also make it easier to reason about practical problems that can be solved with a trigonometric approach, in particular this set of functions makes it clearer on why the definition cyclometric functions exist.
So why this good old approach is not being considered anymore ?
 A: They're not deprecated; they're largely just forgotten.
In the present day, trigonometry is taught in courses for non-mathematically inclined people who have no desire to take a math course except in order to get rewarded with a good grade for obeying and working hard.  They do not suspect that mathematics is a subject in which one derives things logically from other things and that one knows that something is true by understanding its derivation, rather than by being handed a dogma by authorities.  They bring in tuition money.  It's a racket.  Racketeers get paid to pretend that that is education.  If you tell the truth to such (using the term loosely) students, that mathematics is an intellectual endeavor, they complain that other instructors don't require them to know that.
Textbooks written for that kind of audience is where people learn trigonometry nowadays, including many people who actually want to understand mathematics.
It is largely forgotten that there is actually such a thing as an advanced trigonometry book for an audience consisting of mathematicians.  Some of these are of that sort.
