# Factoring trig expressions

This should be simple but I am horrible at math.

Anyways I forgot basic math properties and when I try to work it out in more simple terms I can't make sense of anything. Anyways I have to factor $(\sin x+1)^2 - (\sin x-1)^2$

So I figure that I just do $(\sin x+1)(\sin x+1)$ for the first term and then $(\sin x-1)(\sin x-1)$ for the second I wasn't too sure on the second one but I figured that since the negative sign was outside the expression that I just leave it there, do my math and then apply it. That makes sense to me with what I know of how parentheses work.

Anyways I got the wrong answer and this is where I think it is incorrect, my book does some very strange stuff that I do not understand like adding instead of multiplying the terms.

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Any time you have a difference of squares, $$a^2 - b^2$$ it can be factored as $$a^2-b^2 = (a+b)(a-b).$$

Here you have a difference of squares, with $a=\sin x+1$ and $b=\sin x -1$, so you can factor it this way, and then simplify to see what you get. Note that this factorization is completely independent of the fact that you are dealing with trigonometric expressions; it's just simple algebra.

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Ugh more things to memorize, still haven't gotten the theorem one whatever it is. The one that is a plus or minus square of something. I must be really bad at math because I must have learned this a dozen times by now but I can never remember it. – Adam Jun 15 '11 at 20:56
@Adam: The people answering questions have dealt with similar expressions hundreds, probably thousands, of times. You need to practice, practice, practice. After a while, it will be natural and easy. – André Nicolas Jun 15 '11 at 21:48
But I have done these problems hundreds of times for the last couple years, and I forget them for every single class I am in, I forget everything I have learned no matter what it is and I always just barely get by understand the math well enough to get a C in the class or whatever. I mean I do almost 6 hours of homework a day but 5 hours of that is sitting there incredibly frustrated trying to figure out what to do. – Adam Jun 15 '11 at 21:56
@Adam This isn't something you need to memorize but, rather, recognize. Note that if you use the usual rules of multiplication on $(a + b)(a - b)$ you get the difference of the squares, that is $(a + b)(a - b) = (a+b)(a) - (a+b)b = a^2 + ab -ab - b^2 = a^2 - b^2$. Once you've multiplied this out a few times, say using expressions, instead of just the symbols $a$ and $b$ you should be able to recognize it in different contexts. – ItsNotObvious Jun 15 '11 at 22:47
@Adam: Hopefully this community can help you gain skill and confidence. We'll get you down to 4 hours of homework a day in no time! – The Chaz 2.0 Jun 16 '11 at 3:45

Arturo's answer is the 'slickest' way to get to the answer, but if you wanted to be more pedestrian you could just multiply out the brackets one at a time, and then do the necessary simplifications:

\begin{align} (\sin x + 1)^2 - (\sin x - 1)^2 &= (\sin^2 x + 2\sin x + 1) - (\sin^2 x - 2\sin x + 1) & \textrm{(multiply out the brackets)} \\ &= \sin^2 x + 2\sin x + 1 - \sin^2 x + 2\sin x - 1 & \textrm{(take the negative of the second bracket)} \\ & = 2\sin x + 2\sin x & \textrm{(the }\sin^2x\textrm{ and 1 terms cancel)} \\ & = 4\sin x & \textrm{(collect everything together)} \end{align}

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This is more of an equestrian approach, in my opinion. We'll just have to agree to disagree :)  But seriously, +1 – The Chaz 2.0 Jun 16 '11 at 3:47

It seems to me, from this and many other comments, that you are approaching this (and previous mathematical courses) as subjects to be solely (or mainly) memorized. While it is certainly true that some things need to be memorized (e.g., the multiplication tables, some basic values for sine and cosine, etc), it is completely impractical (not mention nigh impossible) to memorize everything you seem to be trying to memorize. Not only is it very difficult, it is probably why you keep "losing it" after a bit of not using it.

Imagine that you have an exam in which you are required to write out the plot of a novel or movie. One way to do this would certainly be to memorize a description of each chapter (or scene) and then write out these descriptions in the exam. You won't remember what it is about a week later, though. Instead, the way we remember plots of novels and books is understanding the main ideas and mechanics of those plots, and then explaining them in our own words and using our own expressions and ideas.

Mathematics needs to be approached the same way, lest the sheer amount of material overwhelm you (as you are clearly overwhelmed). The key is to recognize and understand the key ideas that are at play, and then use them as organizational principles that allow you to figure out how to do all sorts of related problems. There are lots of things in basic math courses that students think they need to "memorize"; the vast majority of them I've never memorized at all (even when I teach the course)! Instead, I understand the main idea and I figure out the things as I need them on the fly.

In my experience, that is the kind of mindset that makes success in mathematics courses and allows you to retain the important things for later use. Just trying to memorize everything is going to (i) give you a headache; (ii) be extremely hard to do; (iii) make it very difficult to do well; and (iv) make it very easy to forget everything later. There's just no upside to it, even if it is the only approach you have ever used. I know it takes a lot of effort to change approach, but the rewards are worth it; and given how you seem to feel right now, I doubt any change would make you feel worse!

Really, one should be stingy with memorization; as Sherlock Holmes put it, your brain has only a finite amount of capacity, there's no point in filling it with whatever comes your way. Part of the key to success in mathematics is to recognize what needs to be memorized, and what needs to be understood instead. You are trying to approach it by memorizing pretty much everything, and that just doesn't work (as you've discovered).

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That is interesting, I think my method before to learn something was just repetition and replication. I really should take the time to learn everything more thoroughly. – Adam Jun 16 '11 at 3:52
@Adam: Like I said, memorization is important, and so is drill. But drill is done to facilitate use, not to memorize (you drilled with writing simple words not so you could memorize the muscle movements needed to write "mom", but so you could gain facility writing "m"s and "o"s that you can use later). And there are things one does need to memorize. The key really is figuring out what to memorize, and what to understand and be able to recreate as needed. – Arturo Magidin Jun 16 '11 at 3:58