Why did this student think that this verified the identity?

(Thanks Michelle!)

  • I very much doubt that the student knows what is meant by verify.

    • Laura C.

      There’s also the possible confusion of trig identities with each other. Just the concept of squaring functions makes some kids’ heads spin!

      Add in a vague memory of “dividing by stuff” from solving linear and rational equations, and you get…a mental mess.

      A further issue comes from the fact that proofs in general are very difficult if the student’s never seen them before a trig/pre-cal class. The first instinct is to “solve the equation,” even if that doesn’t really work in context.

      So many students have it in their heads that variables mean to solve for x that it’s hard to get them to take a closer look at the problem. I get “x = 14” as the answer to “Simplify 7(4x – 2)” all the time. I think this may be a case of confusing proofs with equations to solve, complicated further with an imperfect understanding of trig functions.

      • It might be interesting to know how the same student would handle a similar, easier problem minus the trig: “Show that x^2y^2 + x^4 = x^2 if x^2 + y^2 = 1.” Too easy? Maybe, but a disturbing number of high school students don’t really understand that a fraction is division, and how to apply that to equations, and this student appears to be one of them. I’ve never taught trig but I imagine fraction and equation solving misconceptions from earlier could really mess students up.

  • A nod is as good as a wink to a blind horse. Apparently the usual rules of solving and manipulating equalities go out the window when trig is involved. It’s possible that the student has some vague recollection of a few of the typical moves made in verifying trig identities and this is what resulted from that. I know that’s not the level of error analysis we want, but I’m not sure that there’s a lot deeper going on here. What do you think, Michael?

  • The student knew she had to divide to cancel something out, but she wasn’t sure what, and apparently forgot her basic algebra. If she had divided every term by sin^2(x), she would have gotten sin^2(x)+cos^2(x)=1, which is an established identity.

    • Laura C.

      It’s reminding me of the “multiply times LCD to solve a rational equation” idea, except A: the student isn’t multiplying everything by the LCD, and B: this isn’t a rational equation.

  • Tiffany Tao

    It’s pretty obvious the kid wasn’t even beginning to try. He was just filling in answers hoping he’d get points. As Celeste said, he or she just dvided to cancel something out, but maybe the kid didn’t understand what’s going on in this verification. If the kid remembers cos^2(x)+sin^2(x)=1, he or she will divide sin^4x to two sin^2x and than do the calculation.