# When do you make use of “≡” when verifying trigonometric identities?

My understanding is that I only use the symbol in lines where I make use of trigonometric identities.

Example: Prove $1+\tan^2x=\sec^2x$

$1+\tan^2x$

$≡1+\frac{\sin^2x}{\cos^2x}$ I make use of the triple bar here because I used a trig identity.

$=\frac{\cos^2x+\sin^2x}{\cos^2x}$ I only make use of the equal sign here since I didn't use any trig identities.

$≡\frac{1}{\cos^2x}$ I used the triple bar again because I used an identity.

$≡\sec^2x$

Is my understanding on its usage correct? Or should I be using the triple bar for all lines?

• It has nothing to do with trigonometric identities, it has to do with other things – Zelos Malum Oct 30 '15 at 13:59
• Yes, I get that. But say for example I'm using this sign when I'm verifying an identity. Am I actually using it right? – boni12345 Oct 30 '15 at 14:01
• The standard use of $\equiv$ is completely different. None of the uses you propose are helpful or likely to be considered correct to the general mathematical reader. – Simon S Oct 30 '15 at 14:01
• This sign is used when something is identically equals to another thing (in your case, it is true for any $x\neq \pi/2+k\pi$. I'd recommand to use $=$ and not $\equiv$, the latter being used in other fields of mathematics for particular purposes. – MoebiusCorzer Oct 30 '15 at 14:02
• No, the tripple line equality sign is commonly used for either definition, this is the case because I define it so, or it has been defiend as such. Or because we're dealing with modulo. – Zelos Malum Oct 30 '15 at 14:02

In this context, I would recommend either:

1. don't use $\equiv$ at all, or
2. only use $\equiv$ to indicate implicit universal quantification.

Let me explain what I mean by (2).

There's this concept of universal quantification that all mathematics is based on. Its really important. If you understand it, math probably makes sense to you. If you don't, it probably doesn't. Unfortunately, this concept usually isn't taught until very late into one's university education, despite that essentially all of high school mathematics secretly uses it all the time! This makes mathematics look way crazier than it really is, which is a real shame.

So, let me try to explain it.

This will have the added benefit that you will know when to use $\equiv,$ in line with recommendation (2). Of course, you can always take recommendation (1) and just forget about $\equiv$ altogether... In any event, understanding this stuff is REALLY important.

Universal quantification.

Suppose we know that $\sin^2 \theta + \cos^2\theta = 1$.

Then obviously, we can deduce that $\sin^2(\theta+1)+\cos^2(\theta+1) = 1$.

(Think about why. This should be intuitively obvious. Don't read on until its obvious.)

Okay, suppose we instead know that $3x=2$. Can we deduce that $3(x+1) = 2$? Of course, we cannot. After all, the first statement is equivalent to $x=3/2$. The second statement is equivalent to $x=1/2$. Obviously, neither implies the other.

So what's really going on here?

Let me explain.

If someone says "we know that $3x=2$," then what they probably mean is exactly what they said. Basically, $x$ is understood to be some fixed but arbitrary real number, and we know that $3$ times that number equals $2$, or in other words that $3x=2$. We can then proceed to find $x$ (if we want).

But if someone says "we know that $\sin^2 \theta + \cos^2\theta = 1$," what they probably mean is: "we know that for each and every real number $\theta$, it holds that $\sin^2 \theta + \cos^2\theta = 1$." Obviously, we cannot proceed to find $\theta$ in this case, because we're talking about each and every possible $\theta$, not some specific $\theta$.

So if we want to be precise, if we want to denote things in a way that makes this difference in meanings clear, then obviously, we have to change our notation a bit.

The statement: "We know $3x=2$" remains unchanged.

But the other statement becomes: We know $$\mathop{\forall}_{\theta:\mathbb{R}}(\sin^2 \theta + \cos^2\theta = 1).$$

Let me explain how to read this. The symbol $\forall$ is verbalized "for all," or "for each," or "for every." A bit of terminology: we call $\forall$ the "universal quantifier." So the above pattern of symbols can be read: "for all $\theta$ in $\mathbb{R}$, it holds that $\sin^2 \theta + \cos^2\theta = 1$."

Often, we simply use words instead of symbols, so $$\mathop{\forall}_{x:X}[\mbox{blah blah}]$$ becomes: "for all $x$ of type $X$, blah blah."

Now please Google the words "function" and "predicate" if you're unfamiliar with these terms, because here comes the tricky part.

If you think about it, you'll see that if we're given a function $f : Y \leftarrow X$ and a predicate $P$ on $Y$, then from the statement $$\mathop{\forall}_{y:Y} P(y),$$ we can deduce $$\mathop{\forall}_{x:X} P(f(x)).$$ This explains how we got from $\sin^2 \theta + \cos^2\theta = 1$ to $\sin^2(\theta+1)+\cos^2(\theta+1) = 1.$ What's really going on is the following. We know that:

$$\mathop{\forall}_{\theta:\mathbb{R}}(\sin^2 \theta + \cos^2\theta = 1).$$

Now write $P(\theta)$ as shorthand for $\sin^2 \theta + \cos^2\theta = 1$. So we know that $$\mathop{\forall}_{\theta:\mathbb{R}}P(\theta)$$

Now define a function $f : \mathbb{R} \leftarrow \mathbb{R}$ as follows:

$$f(\theta) = \theta+1$$

Then we can deduce $$\mathop{\forall}_{\theta:\mathbb{R}}P(f(\theta)).$$

That is:

$$\mathop{\forall}_{\theta:\mathbb{R}}P(\theta+1).$$

That is:

$$\mathop{\forall}_{\theta:\mathbb{R}}(\sin^2 (\theta+1) + \cos^2(\theta+1) = 1).$$

which is what we were trying to show.

This explains why this pattern of reasoning works here, and not with $3x=2$. The notation $3x=2$ isn't shorthand for $$\mathop{\forall}_{x:\mathbb{R}}(3x=2),$$ since that would simply be false.

In summary, if we want our notation to make clear what the "rules of the game" are, then its a good idea to include the symbol $\forall$ or the phrase "for all" in our mathematical writing.

Suppose you want to emphasize that the pattern of symbols $$\sin^2 \theta + \cos^2\theta = 1$$ is really a shorthand for the more correct statement $$\mathop{\forall}_{\theta:\mathbb{R}}(\sin^2 \theta + \cos^2\theta = 1).$$

Then, if you want, you can write $$\sin^2 \theta + \cos^2\theta \equiv 1$$ to emphasize this.

Example.

Let me finish by illustrating how and when to use $=$ versus $\equiv$, in line with recommendation (2).

Proposition. $1+\tan^2x \equiv \sec^2x$ over the real numbers.

Proof. Let $x$ denote a fixed but arbitrary real number. Then the following are equivalent.

1. $1+\tan^2x = \sec^2x$
2. $1+\sin^2 x / \cos^2 x = 1/\cos^2 x$
3. $\cos^2x+\sin^2 x = 1$
4. TRUE

Hence $1+\tan^2x \equiv \sec^2x$ over the real numbers. This completes the proof.

• This is an enjoyable read. :-) – kobe Oct 30 '15 at 16:15
• Good read. Not sure if it matches the level of the question, but a very good read nonetheless! – zahbaz Oct 31 '15 at 5:30
• @zahbaz, thanks. I think a better answer would be longer, and would give an intuitive explanation of the basics of functions and predicates, thereby "reducing" the level of the answer. But I just run out of steam sometimes... – goblin Oct 31 '15 at 5:31
• Why do you use types? Why not $\forall_{\theta\in \mathbb{R}} P(\theta)$? Or, if you insist on using types, wouldn't the Pi-type $\prod_{\theta\colon\mathbb{R}}P(\theta)$ be the standard way to encode universal quantification? You are also the first person I've ever seen who consistently writes $f: Y\leftarrow X$ instead of $f: X\to Y$. That's a lot of unconventional notation for an answer that explains some notational conventions ;) – Andrey Tyukin Oct 31 '15 at 13:01
• @AndreyTyukin, I mean, ZFC is just so weird and quirky and strange, I don't really accept that it's the "standard" foundation for mathematics in any meaningful sense. And sure, it is often claimed that ZFC is "standard," but the material set theory approach is just soooo far from how I view things, I find it impossible to take these sorts of claims altogether seriously. – goblin Oct 31 '15 at 21:35

Unless you are working on some foundational stuff where you have to treat judgemental and propositional equality differently, I think it's mostly matter of taste, and depends on the conventions used in your lecture / in the book that you are reading.

I would read the triple bar here as "equal by definition", whereas "=" would indicate that some identity has been used. So, the first triple bar is correct (I assume that the $\tan$ function has been defined by $\tan(x) :\equiv \sin(x)/\cos(x)$). The usage of the second triple bar is not correct, because you neither use $\sin$ and $\cos$ to define $1$ (the definition $1 :\equiv \sin^2(x) + \cos^2(x)$ is completely nonsensical), nor the other way round (the definition $\sin^2(x) + \cos^2(x) :\equiv 1$ would be even more absurd). The last usage of $\equiv$ is again correct, because $\sec(x) :\equiv 1/\cos(x)$ by definition.

I would rather use $a \overset{\textrm{def}}= b$ if I wanted to emphasize that the two expressions on both sides are equal by definition. Or just leave all the confusing decorative elements out, and use only simple "="-equality signs.

While uses of the triple bar "$\equiv$" are varied, it does have a meaning in connection with distinguishing identities (equations that hold for all unknowns) versus "contingent" equalities (equations that hold for selected values of unknowns).

If an equation has no unknowns, there is no distinction to be drawn. But consider this well-known identity:

$$\sin^2 \theta + \cos^2 \theta \equiv 1 \tag{1}$$

which holds for all angles $\theta$.

Equation (1) might be useful in solving for a particular value of $\theta$, but some additional information would need to be supplied, like this "contingent" equality:

$$\cos 2 \theta = 1/2 \tag{2}$$

The latter equation is not true of all angles, but only for some. Hence we do not consider (2) to be an identity (meaning "identically true").

The triple bar symbol is used to show identity... Here's one way to consider its usage:

$A \equiv B\quad$ The triple bar implies that $A$ and $B$ are identically the same, all the time, for whatever values that may be inputted to them in the specified domain. For example, $1 \equiv 1$. No matter what happens, those two are identical!

$A = B\quad$ On the other hand, using the equals sign states that the left hand side and right hand side are balanced, for at least some value. For example $1=x$ for well, only one value of $x$, which is $x=1$. If no value can be found to ensure equality, then the statement is false.

In practice, outside a few realms of mathematics, you won't often come across a need to use the triple bar $\equiv$ at all. A lot of individuals tend to use $\equiv$ mostly for emphasis when one reaches some notable identity in a proof. The typical equals sign should suffice in nearly all of your cases.

Also, some individuals have noted that the triple bar is used to symbolize definitions. This is true. You might encounter either $\equiv$ or $:=$ or even some other notations to represent a definition. And well, definitions are identical to themselves, so for most intents and purposes, you may as well read $\equiv$ as identical.