He probably doesn't use those trig identities each day anyway. It's basic stuff, you'd think a math professor will be doing research that's slightly more involved than that! Heck, if your research involves lots of trig identities (somehow), you are dumb if you do it by hand, as you'd be wasting time with inevitable mistakes. If you need lots of trig identities done, you use a symbolic math package to do it for you. I'd think there's plenty of profs out there who do quite high level research and would pretty much suck at some of the undergraduate level stuff. I don't remember the trig identities, even if I could prove or derive every one of them in a couple of minutes.

Then there's the problem of profs assuming that things were covered in previous classes which weren't covered. When I got back to math, I had to very quickly memorize a huge number of math facts that I hadn't been expected to memorize, which put me at a distinct disadvantage to most of the other students whose teachers had expected them to memorize them.