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?
[Math] Why memorize trig identities
algebra-precalculusself-learningsoft-questiontrigonometry
Related Solutions
If you're interested in the subject, you should avoid getting trapped into thinking that mathematics consists only of those things you find in the curriculum of a university. That might mean you should look at some popularizations before deciding which things to try to learn. I read David Bergamini's Mathematics as a kid. It gives a different and far more truthful impression of the subject from what you'd get by doing the "bare minimum" in school. Not up to date but I still think it's worthwhile.
I like C. Stanley Ogilvy's Excursions in Geometry. It's amazing how much he can do with so little needed in prerequisite knowledge. Some of it is extraordinarily beautiful. (Follow the link and you'll see that one of the five-star reviews on amazon.com is mine.)
Some of the books published by the Mathematical Association of America are probably worth looking at.
At a more advanced level---say upper-division-undergraduate level--- (Part of the point of some of the comments above, is that you shouldn't only work at a more advanced level. But it's also necessary to do stuff at a more advanced level.) if you like discrete math, maybe Brualdi's textbook on combinatorics and Wilf's Generatingfunctionology. If you like stuff with lots of engineering and scientific applications, maybe Strang's linear algebra book. Very applied. (Here's the difference between "pure" and "applied" mathematicians: the former know about spectral decompositions of real symmetric matrices; the latter know about singular-value decompositions.) Regardless of whether you want "pure" or "applied" stuff, Dym & McKean's book on Fourier series and integrals can teach you something. (It's not very good for learing the analysis background; it's superb for reading about lots of examples of uses of Fourier theory.)
To be continued, possibly in a separate answer later, maybe........
First, bravo to you for taking initiative and immersing yourself in the beauty of learning!
Here are some thoughts for your consideration:
Elementary Linear Algebra [Hardcover] Ron Larson (Author)
Schaum's Outline of Linear Algebra Seymour Lipschutz (Author), Marc Lipson (Author)
Also, try your local university library and see if you find books that suit your needs.
You can also try open course-ware like:
I would also recommend you learn a Computer Algebra System (CAS) like Mathematica or Maple (you can purchase Student Versions) or other free ones (like SAGE or Maxima) because that will really help exploring and learning this topic so much richer!
See: http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems
Lastly, I think you should consider learning proofing methods and here are some books that do a reasonable job at teaching those.
General Proof Strategies
How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) [Paperback] G. Polya (Author)
How to Prove It: A Structured Approach [Paperback] Daniel J. Velleman (Author)
The Nuts and Bolts of Proofs, Third Edition: An Introduction to Mathematical Proofs [Paperback] Antonella Cupillari (Author)
How to Read and Do Proofs: An Introduction to Mathematical Thought Processes [Paperback] Daniel Solow (Author)
Hope that helps and gives you some ideas!
Best Answer
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.