My algebra teacher in college did the first mistake. I still remember it after 21 years: 1. She was reducing some formula, and got it down to cos(a - 0), which

My algebra teacher in college did the first mistake. I still remember it after 21 years: 1. She was reducing some formula, and got it down to cos(a - 0), which is obviously cos(a). The problem is, her cheat sheet had the solution as sin(a). 2. She "expanded" to cos(a - 0) = cos(a) - cos(0) 3. We know cos(0), so this leads us to cos(a) - 1 4. This is where she got stuck. After thinking for a long time she wrote on the board: "(from trigonometry) cos(a) - 1 = sin(a)" 5. Final answer: sin(a) I went up to the board and explained step 2 was wrong and how she should've done it, and I explained that step 4 was wronger, because that would mean that for some angles cos/sin would be outside the [-1, 1] range. She said, "let's try your way then, young man", and doing the expansion on cos(a - 0) got it to cos(a). She then proceed to say I was wrong, because the answer had to be sin(a). I still thank her for teaching me to question authority.