Depends on the equation, honestly. Something like sin² + cos² = 1, should be second nature. Embarrassingly, the sin = opp/hyp is something I always only remember by reciting sohcahtoa. And then you got stuff like sin(x+y) = ... which I just google every single time.

So no, as a mathematician I don't remember many trig equations. Google exists and I make use of that. What matters is not that you remember the equations but that you know they exist and what they mean. If I see for example an integral of cos²(x), I know I need a formula to express this in terms of cos(2x), but I don't remember what formula that is, I just google that. And of course I should know what sine means exactly both in triangles and the unit circle.

It's never a good idea to memorize in math. Rather, you should remember something by using it a lot or by "visualizing" what is going on. I know that cos² + sin² = 1 since I use it every single day. And I know that sin(pi - x) = sin(x) since I just see the unit circle in front of me which makes this relation obvious, I haven't memorized this, nor can I recite it from memory.