Most often they are used for simplifying expressions. For that purpose it's not a matter of deducing what they are, but recognizing them in the expression, and spotting the opportunity to use the identity on the path to eliminating terms.
For that purpose one needs to have them memorized in order to spot the opportunity -on the carefully derived homework problem- in the wild.
If I'm stuck with a trig function I don't know what to do with, I just look on Wikipedia for a trig identity and don't worry about derivations or memorization.
This trick is good for exam settings though! My memory is terrible so I always relied on tricks like this in school
Yes, most high schools spend a few months teaching “trigonometry” as a system for simplifying formulas that was primarily useful historically to save time for human calculators working with pen/paper and a printed lookup table. In the modern world this is entirely anachronistic and the content could be profitably compressed down to a couple weeks, with the saved time spent on something else. The main remaining utility is as rather cumbersome, unpleasant, and unenlightening algebra practice.
In the same sense that doing this work by hand will build depth of understanding, I know that remembering the trig identities came before really understanding their relationships. And that also came far before I could truly understand Euler's formula, radians, etc.