As someone who did trigonometry every week for over a decade, I can say that reinventing it often will really slow you down. There is merit to memorizing the identities (of course, you
should understand them well enough to derive them if you need to).
I was also better at solving problems involving trigonometry than anyone I knew - including my professors.
You could, of course, argue that if you needed to use it every week, you would know all the identities merely by using them so much. Don't assume this is correct, though. One of the reasons I used them so much is because I had memorized them. My colleagues who used them as often as I did and who didn't memorize them did not, in fact, have them burnt into memory after so much usage.
