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.

  1. 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 $\sin^2(\theta)+\cos^2(\theta)=1$.

  2. 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.

  • 1
    $\begingroup$ For what it's worth, I remember somewhat how the tangent sum/difference identities go because of its connection with the addition of velocities formula from special relativity (something I just happened to learn/memorize when I was in middle school and seeing it in public library books, such as by Isaac Asimov), and also from working with the tangent half-angle substitution when calculating indefinite integrals. To get the formula exactly correct, I then try some angles I know the values for (e.g. tangent of 45 degrees is 1 because lines with this inclination have slope 1). $\endgroup$ – Dave L. Renfro Jul 19 '16 at 18:34
  • $\begingroup$ fair enough. Also +1 for Asimov $\endgroup$ – qbert Jul 19 '16 at 18:34
  • $\begingroup$ (continued) Of course, what works for me is probably not going to work for you. My point is that the more you learn, the more connections you'll form between various things that were previously "things to memorize in isolation from other things". That said, I suspect you won't need the tangent formulas. Probably the Pythagorean identities (the main one, and the other two you get by division of the main one), and maybe the double angle formulas for sine and cosine. For example, being able to trade off a product of a sine and a cosine for half a single sine of twice the angle is often useful. $\endgroup$ – Dave L. Renfro Jul 19 '16 at 18:39
  • $\begingroup$ I have very few memorized, but I do have memorized that $\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$ and $\cos(z)=\frac{e^{iz}+e^{-iz}}{2}$, from which most identities can be proven. $\endgroup$ – JMoravitz Jul 19 '16 at 18:57

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.


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