After I went through the steps the student asked in regard to the fourth step (the one I've highlighted above), "Are we really supposed to know that you can factor out a sine squared?" I was a little surprised and immediately blurted out a resounding "Yes." To me that step was seemed like a simple thing I learned in algebra when I was factoring equations to find roots, long before I ever took trigonometry. But it made me realize that that connection between algebra and trig doesn't happen for all students. There's a tendency so see math classes with different names as completely different animals, when in reality they are all interconnected and require knowledge from the previous courses. With this in mind I showed the student that this trig function is similar to x^2+x^2y and I asked him if he would recognize that he could factor out an x^2, so it would look like x^2(1+y). Luckily, this change of perspective illuminated the reasoning behind factoring out a sin^2(x) and he had no trouble understanding how we solved the trig problem.
Students in the tutoring center also have the common misunderstanding that the trig function is being multiplied by x (or theta, etc.). This is an unfortunate misconception because it misses what it means to be a function. Sine, cosine, tangent, and their inverses mean nothing without their inputs--that's why they're called trig functions.
Overall I find that talking to students who don't understand the material right away is extremely helpful because it makes me realize what holes I need to fill as a future teacher. To me math is like breathing, easy and unconscious, but to others it's laborious and they need as many details as possible, especially in seemingly foreign topic like trigonometry. But by understanding these common misconceptions I feel better about helping my future students simplify trig functions. And I believe that these could be simple things to teach in the classroom. As a teacher, I just need to remember to relate new concepts back to old ones and convince the students that there are many similar rules that apply to different functions, even though it may not be obvious at first.
