Should I be able to prove Law of Cosines, Half Angle formula, etc? This is more of a general question then the title suggests, but the laws in the title are what I'm currently studying. I can read the proofs of both and understand them after a while, but I could never produce such a proof unless I committed it to memory which is really my question: should I be concerned about needing to prove something like the Half Angle formula for the sine function on my own is or is it enough to be able to read the proof and understand it?
My long term goals are to read Apostol's calculus or Spivak's but right now I'm just studying Pre-Calc. In the case of Spivak's book I know he gives a proof of why $-a*b = -(a*b)$ for positive $a$ and $b$ and I can follow it but I could never have come up with it on my own. Should I be discouraged by that or does that just mean I need to study harder. Anyways, Cheers!
 A: The mathematical world has billions of proofs which have been worked out over hundreds (thousands?) of years by lots of brilliant mathematicians. That simple thing someone teaches you may have baffled fleets of mathematicians for years before someone came up with a solution. Once it was solved, then it was "simple".
Why would you think you should know how to do the same, by yourself, in a few weeks, when you are studying pre-calc?  You should move forward of course, but you don't have to expect the ridiculous of yourself. 
One way to look at proofs is that there are two general categories:  proofs for which known techniques are effective; and proofs where you have to make something up.
The proofs you cite use known techniques, and the best way to get better at proving that kind of thing is to learn a lot.  If you have a toolbox of known techniques and mathematical knowledge, then you can use those when you are presented with a problem that might fit them.  And the more techniques you have learned, the more problems might fit. 
In the case of the half angle formula, if you could round up 10 things you know about sines and cosines, you might see something in that collection which gives you a clue how to proceed.  If you don't know 10 things, maybe finding the proof for yourself is beyond you at this point.
You have to make something up if either there is a known technique which you are not aware of, or there is no known technique.  Again, the more math you know, and the broader and deeper your knowledge, the better your chances of coming up with a good idea.
The need to think up new approaches, or to make previously unknown connections among several known things is very creative, and in that sense mathematics is an art.  The best people seem to operate by intuition; but that intuition is really years of serious experience with the subject, along with an individual way of thinking about things.
Still, there are a couple of good tips for getting started on a problem. The first, which students must listen to, is that you have to make sure you know the definitions of all the terms.  No guessing.  You can't solve any problems unless you know what the question is.
The second step is to take a divide and conquer approach.  This means you chip a piece off the problem and work on that.  Often that means proving whatever it is for a simple case.  For example, suppose you are given a problem about a curve, and you think you could solve it if only the curve were a straight line.  Fine, prove that much. Now say -- well what if the curve is nearly a straight line?  Would its deviation from straightness invalidate my proof, or could I find an argument that says the slight deviation doesn't matter?  Of course, you have to define what you mean by "nearly straight".  Then maybe you move on to curves that aren't nearly straight.
Or you could go about your chipping this way: can I solve this if the curve is y = $x^2$?  What about if the curve is y = sin(x)?  If you can, prove it for those cases.
By the time you've worked a number of examples of that kind, you may be getting an idea about the general nature of things, and can perhaps expand your proof of a simple case into a proof of the general case.   
And if not, join the crowd.  Everyone mathematician is working on problems he/she does not know how to solve.   
A: All the trig identities become easy to remember and easy to prove once you learn the connection between complex exponentiation and trig functions. 
For me personally, I did not learn about that connection until sometime in college, after or during a calculus course on sequences and series. Until then, I was aware that trig identities existed and was comfortable looking them up when I thought they might help. 
