Proving identities – what’s your philosophy?

What happens in your classroom when you give students the following task?

Prove $1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$.

Sometimes the command is Verify or Show instead of Prove, but the intent is the same.

Two non-examples

Here are two ways that a student might work the problem.

Method 1

$1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$

$1+\sec{\theta}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$

$(\sec{\theta}-1)(1+\sec{\theta})=\tan^2{\theta}$

$\sec^2{\theta}-1=\tan^2{\theta}$

$\tan^2{\theta}=\tan^2{\theta}$

Method 2

$1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$

$1+\sec{\theta}=\frac{\sec^2{\theta}-1}{\sec{\theta}-1}$

$1+\sec{\theta}=\frac{(\sec{\theta}-1)(\sec{\theta}+1)}{\sec{\theta}-1}$

$\sec{\theta}+1=\sec{\theta}+1$

How do you feel about these methods? In my opinion, both methods represent a fundamental misunderstanding of the prompt. Method 1 is especially grotesque, but Method 2 also leaves a lot to be desired. Let me explain. And if you think the above methods are perfectly fine, please be patient and hear me out.

This is the crux of the issue:

The prompt was to prove the statement. But if the first line of our work is the very thing we’re out to prove, then we are already assuming the thing we want to prove. We’re Begging the Question.

It’s as if someone demands,

“Prove Statement X, please!”

and we reply,

“Well, let’s first start by assuming that Statement X is true.”

This is nonsense.

What went wrong?

So what is the proper way to engage this proof? Let’s roll back a bit.

The error in these approaches seems to stem from a desire to perform algebraic operations on both sides of an equation in the same way that you might if you were solving an equation.

When we “do algebra” and write Equation B below another Equation A without any words, we always mean that Equation A implies Equation B. That is, when we write

Equation A

Equation B

Equation C

etc…

we mean that Equation C follows from Equation B, which follows from Equation A.

Some might claim that each line should be equivalent to the last. But, again, when we “do algebra” by performing algebraic manipulations to both sides of an equation to transform it from equation A into equation B, we always mean $A\Rightarrow B$, we don’t mean $A\iff B$. Take, for example, the following algebra which results in an extraneous solution:

$\sqrt{x+2}=x$

$(\sqrt{x+2})^2=x^2$

$x+2=x^2$

$0=x^2-x-2$

$0=(x-2)(x+1)$

$x=2 \text{ or } x=-1$

In this example, each line follows from the previous, however reversing the logic doesn’t work. But we accept that this is the usual way we do algebra ($A\Rightarrow B\Rightarrow C\Rightarrow \cdots$). Here the last line doesn’t hold because only one solution satisfies the original equation ($x=2$). Remember that our logic is still flawless, though. Our logic just says that IF $\sqrt{x+2}=x$ for a given $x$THEN $(\sqrt{x+2})^2=x^2$.

As we move through the algebra line by line, we either preserve the solution set or increase its size. In the case above, the solution set for the original equation is {2}, and as we go to line 2 and beyond, the solution set is {2,-1}.

For more, James Tanton has a nice article about extraneous solutions and why they arise, which I highly recommend.

So if this is the universal way we interpret algebraic work, which is what I argue, then it is wrong to construct an argument of the form $A\Rightarrow B\Rightarrow C$ in order to prove statement A is true from premise C. The argument begs the question.

Both Method 1 and Method 2 make this mistake.

How does a proof go again?

I want to actually make a more general statement. The argument I gave above regarding how we “do algebra” is actually how we present any sort of deductive argument. We always present such an argument in order, where later statements are supported by earlier statements.

ANY time we see a sequence of statements (not just equations) A, B, C that is being put forward as a proof, if logical connectives are missing, the mathematical community agrees that “$\Rightarrow$” is the missing logical connection.

That is, if we see the proof A,B,C as a proof of statement C from premise A, we assume that the argument really means $A\Rightarrow B\Rightarrow C$.

This is usually the interpretation in the typical two-column proof, as well. We just provide the next step with a supporting theorem/definition/axiom, but we don’t also go out of our way to say “oh, and line #7 follows from the previous lines.”

Example: Given a non-empty set $E$ with lower bound $a$ and upper bound $b$, show that $a\leq b$.

1. $E$ is non-empty and $a$ and $b$ are lower and upper bounds for $E$. (given)
2. Set $E$ contains at least one element $x$. (definition of non-empty)
3. $a\leq x$ and $x\leq b$. (definitions of lower and upper bound)
4. $a\leq b$. (transitive property of inequality)

Notice I never say that one line follows from the next. And also notice that it would be a mistake to interpret the logical connectives as biconditional.

The path of righteousness

I encourage my students to work with only ONE side of the expression and manipulate it independently, in its own little dark box, and when it comes out into the light, if it looks the same as the other side, you’ve proved the equivalence of the expressions.

For example, to show that $\log\left(\frac{1}{t-2}\right)-\log\left(\frac{10}{t}\right)=-1+\log\left(\frac{t}{t-2}\right)$ for $t>2$, I would expect this kind of work for “full credit”:

$\text{LHS }=\log\left(\frac{1}{t-2}\right)-\log\left(\frac{10}{t}\right)$

$=-\log(t-2)-\log(10)+\log(t)$

$=-\log(10)+\log(t)-\log(t-2)$

$= -1 + \log\left(\frac{t}{t-2}\right)$

$=\text{ RHS}$

Interestingly, I WOULD also accept an argument of the form $A\iff B\iff C$ as justification for conclusion A from premise C, but I would want a student to say “A is true if and only if B is true, which is true if and only if C is true.” Even though it provides a valid proof, I discourage students from using this somewhat cumbersome construction.

So let’s return to the original problem and show a few ways a student could do it correctly.

Three examples

Method A – A direct proof by manipulating only one side

$\text{LHS}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$

$=\frac{\sec^2{\theta}-1}{\sec{\theta}-1}$

$=\frac{(\sec{\theta}-1)(\sec{\theta}+1)}{\sec{\theta}-1}$

$=\sec{\theta}+1$

$=\text{RHS}$

Method B – A proof starting with a known equality

$\tan^2{\theta}=\tan^2{\theta}$

$\sec^2{\theta}-1=\tan^2{\theta}$

$(\sec{\theta}-1)(1+\sec{\theta})=\tan^2{\theta}$

$1+\sec{\theta}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$

$1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$

Method C – Carefully specifying biconditional implications

$1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$

$\text{if and only if}$

$1+\sec{\theta}=\frac{\tan^2{\theta}}{\sec{\theta}-1}$

$\text{if and only if}$

$(\sec{\theta}-1)(1+\sec{\theta})=\tan^2{\theta}$

$\text{if and only if}$

$\sec^2{\theta}-1=\tan^2{\theta}$

$\text{if and only if}$

$\tan^2{\theta}=\tan^2{\theta}$

While all of these are now technically correct, I think we all prefer Method A. The other methods are cool too. But please, please, promise me you won’t use Methods 1 or 2 which I presented in my introduction.

In conclusion

Some might argue that the heavy criticism I’ve leveled against Methods 1 and 2 is nitpicking. But I disagree. This kind of careful reasoning is exactly the business of mathematicians. It’s not good enough to just produce “answers,” our job is to produce good reasoning. Mathematics, remember, is a sense-making discipline.

Thanks for staying with me to the end of this long-winded post. Can you tell I’ve had this conversation with a lot of students over the last ten years?

Further reading

1. Dave Richeson has a similar rant with a similar thesis here.
2. This article was originally inspired by this recent post on Patrick Honner’s blog. A bunch of us fought about this topic in the comments, and in the end, Patrick encouraged me to write my own post on the subject. So here I am. Thanks for pushing me in the right direction, Mr. Honner!

12 thoughts on “Proving identities – what’s your philosophy?”

1. There are a lot of things going on here: proving trigonometric identities, the nature of proof, solving algebraic equations, etc.

I completely agree with you that proofs should be held to a high standard. If the prompt is to prove an algebraic or trigonometric fact, then it is simply incoherent to begin with the statement that you are trying to prove. Without some kind of qualification, like “suppose”, you have immediately placed your audience in the awkward position of having no idea what is going on. If I purport to prove that, say, “Congressional term limits are anti-democratic” and begin my proof with the statement that “Congressional term limits are anti-democratic”, it is worse than awkward, it is just incoherent.

So if we want students to be able to read and understand proofs, to compose their own proofs,and to engage in mathematical communication with each other via proof, then we should definitely insist that the proofs be coherent.

But are the traditional problems that ask students to prove various trig identities really proofs at all? Why are we asking students to prove trig identities in the first place? I would argue that the entire value of this exercise is to get students used to manipulating trigonometric expressions based on simpler known identities. Students must learn their identities, so we give them this practice. If they become adept at this manipulation, then they will be able to fluently compose their own substitutions when needed to solve a trigonometric equation or antidifferentiate or one of the other “natural” situations in which one needs to know their trig identities.

My point is that the task of “proving trig identities” is not really a proof task. We might as well just say: transform the expression on the left to the expression on the right via a sequence of equality-preserving algebraic and trigonometric operations. This is the skill we want them to have. Instead, we just say “prove this identity”, Then we must awkwardly add special class rules that “one can only manipulate one side of the expression at a time” which seems arbitrary and mean.

Methods A, B, and C are equally good as proofs, since all three do indeed prove the identity. But only method A employs the skill that we are trying to build in the first place.

• I will buy this !

• However we are also attempting to make students aware of potential pitfalls in the logical steps they take e.g. that division by a trig expression has to specifically exclude cases when that expression has value zero or that squaring may introduce additional values that may not be satisfied etc. So trigonometrical transformation I would suggest is not the only skill practiced in these type of examples.
May I add that this article is an excellent and clear exposition and should help many teachers through this potential minefield.

• Great points, Will.

Yes, I’ll acknowledge that I was trying to address more points in this post than I should have, biting off more than I can chew, trying to say too much at one time, making multiple claims, etc. I probably should have made a narrower argument or turned this into a series of posts.

To your other point, you’re right that trig identities are a little bit of an artificial place to practice writing proofs. Maybe my opening example shouldn’t have been a trig identity, because really my views here extend to basically all proof tasks in math. I do provide some other non-trig examples in the post. One solution would be to stop using trig identities as a vehicle for teaching proof. I think you would have reason to do this, especially since the practical purpose of this is extremely limited.

My preferred solution would be to *saturate* the curriculum with opportunities for proof. Too many teachers treat proof as a thing you do in isolated places like Geometry or Trig Identities or when you get to that Induction lesson in Precal. Mathematics is a sense-making discipline, and proof should undergird our activities every day in math class. Clear thinking should always be center stage!

It’s an honor to have a logician such as yourself stop by, Will. I should get you to weigh in on my previous post, too, which was all about proof. When it comes to the finer points in logic, I might be out of my depth. So I’m grateful for your comments, and I’m glad to have you as a colleague!

• I would agree about the essential nature of “proof” especially if students are engaged in investigative work. Pattern spotting => formulating conjectures => proving.
Since proof is by far the most difficult and deep activity at least teachers should constantly be advising their students of the need for proof and especially the term “counter-example” and what this means for the status of a conjecture. All this (handled well) can make a huge difference to motivation/enjoyment etc

2. None of the proofs offered pass my muster for proofs that I expect for college-level mathematics majors. I expect them to say something about the domain of rational, square root, and trig functions, “Assuming that theta is not pi/2 …” or “assuming x-2 > 0.”

• True. We should be clear about domain restrictions, especially if one expression has a different domain restriction than the other expression. Once the domain restrictions are made clear, will that meet the level of rigor you would expect?

The definition I’ve often heard for Identity is this: “An identity is an equation that holds for all values for which both expressions are defined.” That definition might not be universally adopted by mathematicians, though, so any domain restrictions should still be made clear, as you suggest.

By the way, I did specify the domain on the log problem ($t>2$). The domain restriction carries through the proof, so do you think that a reference to this domain restriction needs to be addressed elsewhere?

• Domain restrictions can be implicit. For example, “For the rest of this course, we will be restricting ourselves to real-valued functions of real variables.” That does it. I’m not being draconian, only clear.

I don’t like lists of naked equations, though, unless the steps are “just algebra” in a course in which algebra is not the focus. In Algebra 1, I expect English like “by factoring the trinomial,” but in Probability I will assume “by looking it up in a cumulative distribution table for the normal pdf” without further explanation, without actually saying so.

Mathematician E. H. Moore — as quoted by Morris Kline in his book Mathematics: The Loss of Certainty — famously says, “Sufficient unto the day is the rigor thereof.”

3. I like your post, John, but I don’t have anything substantive to add. Where’s the “Like” button? 😉

• Thanks for stopping by, Anthony! You actually can click the Like button below the post if you really want. I’m not sure it’s a very important feature of WordPress, though, since I think few people use it 😊.

4. GREAT DISCUSSION! I’m glad I stumbled upon this…

These types of identities pop their heads up in introductory calculus and precalculus, and it seems as educators (that usually had a lot of math classes) we have been trained in logic and reasoning, so requesting that students prove or verify a statement (especially in this algebraic manipulation setting) seems like a straight forward task that needs little to no explanation. But, in my personal education logic and reasoning was something I never gave much thought to (especially in a calculus type class) and it wasn’t until later in my education that I learned (and fell in love with) logic, as we use the word…
So after that rant, which I apologize for… The question I raise to many of my cohort is: “Should we teach a ‘logic’ course before we expect students to ‘prove’ things or even comprehend conditional statements as many of the theorems in calculus involve?”

Or (as it feels like it is handled now) do we treat logic like it is something born to each person, that has little to nothing said in the classroom?

[I’m just a low level graduate student at a state school, and I’m sorry if I missed something in your post or others comments that answered this, or if this comment is offensive in any way to professionals as yourselves]

• I think logic should be taught as part of every math course. Geometry is often the first course where students encounter logic in a formal way, but that’s only because Geometry is where students first encounter proof for the first time. This is an unfortunate state of affairs.

My vision, shared by most math teachers, I would hope, is that proof would be a central part of every math course. Mathematics is different than other disciplines and the thing that makes it different is that mathematics is a sense-making discipline. It’s only in math class that you don’t have to answer to authorities–you can verify results yourself. And, in a truly ideal world, a student would encounter no unjustified statement in math class.

If math class isn’t logical and doesn’t make sense, what exactly is math class offering? This is our entire business, I say.