I love this trig identity. I could imagine a student believing it for the wrong reason, a grader counting it wrong for the wrong reason, and a teacher counting it right for the right reason.
sin(x – y) sin(x + y) = (sin(x) – sin(y)) (sin(x) + sin(y))
Someone manipulating symbols unknowingly might think this is obviously true: of course you can replace sin(x + y) with sin(x) + sin(y) and replace sin(x – y) with sin(x) – sin(y). All the world is linear.
Someone with a little more experience would say that this identity obviously cannot be true. After all, sin(x ± y) clearly does not equal sin(x) ± sin(y).
But someone with a little more patience might get a pencil and paper and work out that it indeed is true. Even though naive symbol manipulation would be wrong-headed, in this case it happens to lead you to the right result.
For more examples of a novice and an expert agreeing but someone in between disagreeing, see Coming full circle.
I have never seen this before. So cool!
Regarding coming full circle, the Zen (Chan) tradition has some koans that describe this process as a natural part of seeing more clearly. If I recall correctly, in one of them two monks are arguing about whether a nearby mountain really exists or not. On the one hand, it clearly exists to the unenlightened mind. But Buddhist teaching says that the world of the senses is illusion. The master comments, “First there is a mountain, then there is no mountain, then there is a mountain again.”
I think about this process a lot in regards to education. Often it seems to show up with exceptions that require a sophisticated understanding. First there are no rules, then there are absolute rules, then there are exceptions which disprove absolute rules which depending on how you view rules is like being back to no rules.
Also related in my mind is a process of education which can be viewed as temporarily lying to bridge the gap between naïveté and sophistication. Take how integration in calculus is taught. Usually the integral is taught as the area under a curve, and it is shown that in the limit dividing the area under the curve into ever smaller rectangles and adding their areas produces the desired result. Later it is shown that this process can fail but an integral still exists, leading to a more sophisticated understanding of integration.
A related phenomenon is seen in stage magic and in some forms of fraud, where a clearly verifiable event is attributed to the wrong cause. The evidence seems to support the attribution.
That trig identity is like a magic trick!
Wow. That was new to me too!
This trig identity actually puts me in mind of this: http://en.wikipedia.org/wiki/Freshman%27s_dream ,
The Freshman’s Dream; where a rather naive freshman writes down (x+y)^n=x^n+y^n. Turns out that this holds true for a special case in ring theory!
I like using student errors as a way to generate interesting questions. When a student says something that isn’t true in general, it’s interesting to see whether there is another context in which it is true. See, for example, the Freshman’s dream.
einzztein: Looks like your comment came in as I was writing mine.
This is a very cool identity. I couldn’t resist trying to generalize it a bit.. I *think* that the only C2 functions satisfying an identity of this kind are either of the form f(x) = A x or of the form f(x) = A sin(kx),
for constants k and A.
It reminds me of the fraction 16/64, where you can cancel the 6s and get 1/4. Even though the reasoning is wrong, the conclusion is right.
Modern age proof: true, Wolfram says:
http://www.wolframalpha.com/input/?i=sin%28x+%E2%80%93+y%29+sin%28x+%2B+y%29+%3D+%28sin%28x%29+%E2%80%93+sin%28y%29%29+%28sin%28x%29+%2B+sin%28y%29%29