The Philosophy of Proving Trig Identities

[romolo]

I was wondering if anyone could help me out about trig identities. I'm a HS trig teacher and I'm going "by the books" and instructing them to manipulate the left and right sides of the identity independently of each other. They are not to treat it like an equation, e.g. no moving terms from one side to the other by way of adding etc, no squaring or rooting each side, no cross multiplying, etc. The rationale, I guess, is that we don't yet know whether it is a valid equality so the equals sign should have a little question mark above it until the two sides are identical.

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.

 

[slider142]

If they are experienced enough, you can let them use the same algebra that is valid for manipulating inequalities, keeping in mind that the specific inequality is unknown (or whether it is actually an equality, for which the same algebra is valid).
However, a less questionable method of presenting the same proof would have been to pretend it's an equality as the student did until you arrived at a real equality, and then turn it around and present the last line as the first line of the proof and the first line as the last line. this would allow him to catch any errors in reasoning that may have inadvertently been made.
It will also prepare them for epsilon-delta proofs of calculus, which usually get started on a scratch pad working backwards from epsilon to delta, then presenting the forwards approach cleanly from "let delta be ..." covering your tracks in the sand.

 

[Ben Niehoff]

My opinion is that to disallow perfectly valid mathematical steps will serve only to confuse the students and make them think that mathematics is haphazard and arbitrary.

 

[Citan Uzuki]

The problem here isn't whether they manipulate one side of the equation or both. The problem is one of logical validity. When a student is asked to verify some trigonometric identity, they are being asked to prove that it is true. The algebraic manipulations that they produce in response to such a question are a logical argument that the identity in question really is true, albeit one presented in algebraic shorthand rather than English words.

Now, when a student presents a response like the one in the example you gave, the argument he is implicitly making is:

Suppose that cos² x - sin² x = 2 cos² x - 1. Adding 1 to both sides, we have that 1 + cos² x - sin² x = 2 cos² x. Subtracting cos x from both sides, we have that 1 - sin² x = cos² x. Finally, applying the Pythagorean theorem to the left side yields that cos² x = cos² x, which is true. Therefore, cos² x - sin² x = 2 cos² x - 1. Q.E.D.

Leaving out the algebraic manipulations and just focusing on the basic structure of the argument, the student's argument is:

If cos² x - sin² x = 2 cos² x - 1, then cos² x = cos² x, which is true. Therefore, cos² x - sin² x = 2 cos² x - 1. Q.E.D.

Or in symbolic form:

If A then B
B
------------
A

I'm sure you see the problem here. The structure of the argument is NOT valid. It is the formal fallacy of affirmation of the consequent. That this argument is invalid may be seen by replacing A and B with other statements. For instance, we might let A be "I live in Massachusetts" and B be "I live in the United States of America." It is undoubtedly true that A implies B, since Massachusetts is in the U.S., and I do in fact live in the U.S., but the conclusion, that I live in Massachusetts, is not true. I live in Utah.

In general, if a student starts with the identity they are trying to prove is true, and algebraically manipulates both sides to produce an obviously true equation, they are giving an argument like the one produced by your student, which relies on a formal fallacy. That is why such a method of "verifying" trig identities cannot be allowed. However, that does not mean that the only valid method of proving an identity is to start with one side of the equation and transform it to the other. For instance, consider the following argument:

Obviously, cos² x = cos² x. Applying the Pythagorean theorem to the left side yields that 1 - sin² x = cos² x. Adding cos² x to both sides yields 1 + cos² x - sin² x = 2 cos² x. Finally, subtracting 1 from both sides yields that cos² x - sin² x = 2 cos² x - 1. Q.E.D.

This argument has the structure:

B
If B then A
-----------
A

Which IS perfectly valid, even though it involves manipulating two sides of an equation. The important thing is that we are arguing from premises we know to be true to the conclusion that we want to prove, and not from the conclusion we want to prove to premises we know to be true.

Now, this doesn't mean that starting with the conclusion and manipulating both sides is useless. As slider142 notes, in many cases an invalid proof which proceeds from conclusion to premises can be turned into a valid proof simply by reversing all the steps (which is pretty much what I did with your student's argument). And in many cases, it is actually easier to find a path from the conclusion to the premises and reverse it than it is to find a path from premises to conclusion in a more direct fashion. As such, trying to work backwards can be a useful method for discovering proofs. But it needs to be emphasized that:

• Working backwards is only a method for finding proofs, and cannot constitute a proof itself. Only the work from the premises to the conclusion counts as a proof.
  
• Working backwards is only useful if the steps can actually be reversed at the end. If a non-invertible step such as squaring both sides is used, this cannot be reversed, and assuming it can may lead you to present invalid proofs or worse, to incorrectly believe that some false equation is true (e.g. someone who forgets that squaring both sides is not invertible might conclude that x-y = y-x, because they both have the same square).
  
• Working backwards is NOT the only way to construct a proof. You should spend at least as much time trying to work forwards as backwards.

Now, as for what you should allow them to do as a teacher -- I would recommend that in general, you should allow them to prove these identities using any valid argument they like. There's no reason why they should be restricted to just manipulating one side of the equation at a time. But invalid arguments, such as the one presented by your student, should not be accepted. I would also recommend that you spend some time talking about exactly what constitutes a valid proof, and why proofs such as the one your student presented are invalid. Also spend some time discussing the circumstances in which such invalid proofs can be reversed to yield valid proofs, and how it relates to the problem of extraneous solutions in algebra.

(Briefly, the relation is this: The process of solving an equation in algebra amounts to manipulating an equation such as F(x) = G(x) (where F and G are expressions involving x) into the form x = C, where C is some expression not involving x. This amounts to producing a proof that "if F(x) = G(x), then x=C" -- i.e. that C is the only possible solution to the equation. However, such a proof is no guarantee that "if x=C, then F(x) = G(x)" -- i.e. that C actually IS a solution to the equation. Now, if all of the operations used to produce the proof that "if F(x) = G(x), then x=C" are invertible, then that proof can be reversed to show that "if x=C, then F(x) = G(x)" and that reversed proof guarantees that C really is a solution of the original equation. However, if non-invertible operations were involved in producing the original proof, then the proof cannot be reversed, and so you have no guarantee that C is actually a solution of the equation, so you really do have to check. Thus, solving an equation may produce extraneous solutions only if the proof cannot be reversed to show the solution x=C implies the original equation, which occurs only if non-invertible operations were used in deriving the solution.)

 

[m00npirate]

I was taught to prove trig identities by only manipulating one side, which at the time I felt was silly for the reasons you all have mentioned, but I think it does serve a purpose. When doing calc you often have integrals of trig functions that do not immediately appear "nice," but through manipulation you can turn them into something more manageable. Proving these equalities using only one side of the equation (at least for me) helped me to be able to see and make these substitutions more easily.