This is a bit of a two part question. I also have read some of the related questions, but I think mine is different as whether they can be derived quickly, rather than whether they can be derived, is of essence.
Are there some trig identities really worth memorizing for things like the math subject GRE. If so what are they? I am talking about the nasty $\tan(\alpha+\beta)$ formulas, not basic stuff like double angle and variants of $\sin^2(\theta)+\cos^2(\theta)=1$.
Of the ones worth knowing off hand, which can be quickly derived using complex analysis? I tried with the tangent identities and those got painful.
When I took the math GRE the only trig identities I had memorized were $\sin^2\theta + \cos^2\theta = 1$ and $\tan^2\theta + 1 = \sec^2\theta$. I don't recall whether I actually made any use of them.