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.