This looks to me like a “I know I want 1/cos^4 so let’s just cross off or throw away everything else” kind of thing.

More emphasis on factoring and/or common denominators would probably help this student be more likely to start down the right path. But it’s good that at least they have the destination in mind!

Jim Doherty

My first thought is that the student has some vague memory of dividing fractions by reciprocating. If I vaguely remember that I can ‘flip’ a fraction – but I am not sure when/why – then it is tempting to do that here. The next mystery would be why the addition sign in front of this multiplication disappeared. Don’t know how early on in the trig identity unit this exercise is, but I would actually be more discouraged by the fact that the student immediately went to basic sine and cosine representations before thinking about factoring and the pythagorean identities. This kind of behavior here indicates that the student is operating at a very fundamental level in dealing with these functions. It is almost as if tangent and secant are not legitimate functions themselves, just disguises for cosine and sine.

Jim’s first thought makes a lot of sense to me. I always used to wonder why we do so much of this kind of identity-proving in trig classes — most of the identities are utterly useless — but eventually I decided the point was to identify lurking holes in their algebraic manipulation skills without making it seem like we were just doing another algebra unit reviewing things they thought they already knew. Here we can see that this student probably needs more work with fractions, specifically common denominators, and possibly some more work with the differences between addition, multiplication, and division.

I somewhat agree with the second comment about tangent function, but I have to admit that I don’t really think of secant as a legitimate function on its own. I will essentially always do what this student did (except in cases like this one where I see a very clear path to a simple Pythagorean theorem). For instance, it’s only recently that I’ve learned things like for any triangle, tan A tan B tan C = tan A + tan B + tan C, mostly because I only ever really think about sines and cosines.

I think it’s a perfectly good strategy to convert everything to sine and cosine unless you see an easy way to get the answer you want, so I wouldn’t be at all discouraged by seeing that behavior here.

John Fontaine

I agree with second post as well with regard to dividing and using a reciprocal. However, looking at this post reminds me of how many students just make up steps and algebra to “get it”. I often question the necessity of proving statements using identities and wonder what others think.

Mickey

Oh yeah…geometry & trigonometry sure came in handy when I was preparing client tax returns and taking the bar exam. Oh, joy!!!