- #1

romolo

- 7

- 0

I've always found that argument a little soft on logic. If I'm comparing two expressions and would find it convenient to add 15, say, to both of them, surely I have not changed their relationship either from equal to not equal or vice versa. Likewise, if I were to double each expression, I don't see any problem there either.

There might be an issue with multiplying variable expressions (as opposed to constants) on both sides, in that variable expressions may sometime equal zero or be undefined. Also, squaring both sides may introduce extraneous solutions and square rooting both sides might throw us into imaginary numbers. So maybe I'm comfortable banning these actions. (However, I would like to see an example of a "false" equation that, when using these inappropriate methods, results in the erroneous conclusion that the two sides are equal.)

Maybe this is all too theortical. Here's a student's answer to a verification problem that got me thinking of posting this.

The problem:

Verify cos^2(x) - sin^2(x) = 2cos^2(x) - 1

The work:

Step 1 -> add 1 to both sides -> 1 + cos^2(x) - sin^2(x) = 2cos^2(x)

Step 2 -> subtract cos^2(x) -> 1 - sin^2(x) = cos^2(x)

Step 3 -> Pythagorean Identity -> cos^2(x) = cos^2(x)

(I suppose step 3 could have been to add sin^2(x) to both sides, too.)

So here's my question. Do I allow my students to solve identities this way? Notice that in the example above, there was no "inappropriate" steps except adding and subtracting, so the concerns mentioned above are moot. I can think of no reason (apart from them going against my instruction) of disallowing it.