Why memorize trig identities? I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my own. My dad wants me to memorize about 30 trig identities (though some of them are very similar) since I'm missing trig. I've gone through and proved all of them, but memorizing them seems like a waste of effort. My dad is a physicist, so he is good at math, but I think he may be wrong here. Can't one just use deMoivre's theorem to get around memorizing the identities?
 A: Usually, yes, though I prefer Euler's identity. Pretty much every trig identity can be derived from 
$$e^{ix}=\cos(x)+i\sin(x).$$
However, it is useful to memorize some of the common ones because they will help you a lot in calculus and beyond to quickly identify when an expression can be simplified. I would start with memorizing the angle addition formulas. From there you can quickly derive the double and half angle formulas as well as some others.
A: Let me first note that physicists may be better people to ask this question than mathematicians are.
I think it's worth remembering a few, and knowing how to rederive the others.
The important ones to remember initially are: 
(a) The definitions of $\tan$, $\cot$, $\sec$, $\csc$, in terms of $\sin$ and $\cos$, as well as the identity $\cot x = 1/(\tan x)$. 
(b) The Pythagorean identity $\sin^2 x + \cos^2 x = 1$, and the corresponding ones relating $\tan$ and $\sec$, then $\cot$ and $\csc$.
(c) The reduction formulas involving trigonometric functions of $-x$, $\pi/2 - x$, $\pi + x$, $2\pi + x$ (and especially the periods of $\sin, \cos, \tan$). These, as well as the geometric reasons for them, should be learned independently of (d), focusing mainly on $\sin$, $\cos$ and $\tan$.
(d) the angle-sum formulas for $\sin$ and $\cos$.
Eventually, from repeated use, you will probably learn the angle-difference formulas for $\sin$ and $\cos$, the double-angle formulas for $\sin$ and $\cos$, the formulas for $\sin^2 x$ and $\cos^2 x$, perhaps the angle-sum and angle-difference formulas for $\tan$. I have never learned the product-to-sum and sum-to-product formulas and re-derive them whenever I need them, though some people might disagree with this approach.
Overall then, you should learn the most important ones first, and then through practice your list of memorized identities will start to look more and more like the one your dad wants you to learn. 
It is better to use complex numbers for anything involving $3x$, $4x$, etc., and cubes or higher powers of $\cos$ and $\sin$.
Sometimes in calculus, depending on the level you learn it at, it's important to know $\sin x$ and $\cos x$ in terms of $\tan x/2$. There is a nice geometric way to do this by considering $\tan x/2$ as the slope of the line $l$ from $(-1,0)$ to $A = (\cos x,\sin x)$. $A$ can be found as the point of intersection of $l$ and the perpendicular to $l$ passing through $(1,0)$.
A: In favor of rote memorization, unless all your exams are open-notes
you will find that it is very awkward when you have $n$ minutes left to
complete $m$ problems and you are busy re-discovering each trig identity
that you need to use as you go along.
Working out the identity to use in one part of the problem can easily take
longer than the time allotted to solve the entire problem.
For other purposes, such as solving real-life math problems
(where usually you are in a work environment that provides access to
reference books, or even better, to the Internet),
completely memorizing all of those formulas by rote is probably unnecessary.
What is useful is to be aware that there are identities to deal with
certain trigonometric expressions that you may encounter,
as a prompt for you either to derive them or to look them up.
If you encounter certain identities very frequently,
the time you might spend deriving them or even looking them up on every
occasion is wasteful, but in those cases it it likely you will find
that you have memorized the formula after using it some number of times
without consciously trying to memorize it.
Putting the effort in to know these identities
(being able to write them out from memory, but also knowing
how they can be derived)
is a good investment at this time.
But there are options in exactly how you remember the identities;
if you cannot simply write out the formula by rote, but you can doodle
a small diagram in a couple of seconds (perhaps illustrating part
of the derivation of the formula), then write out that formula with that aid,
I think that is good enough.
A: To be honest, the only trig identities you really need are the definitions of the 6 trig functions, and Euler/De Moivre. You can prove almost all the trig identities from $e^{ix}$. Deriving an identity is much easier than just rote memorization, and with repeated deriviation, it eventually becomes memorized. 
