Is $\tan\theta\cos\theta=\sin\theta$ an identity?

Question

A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like "$f=g$" is regarded as an "identity" when one of their domains is a proper subset of the other, and the two functions coincide on the smaller domain. Examples of such equalities are abundant in high school mathematics exercises, e.g. $\frac{x^2-1}{x-1}=x+1,\ e^{\log x}=x$ etc.. Often, the domains are not specified in the exercises.

A somewhat similar but subtly different case is when both functions are defined on the same domain but whether they are equal depends on the exact domain. For instance, $e^{x+y}\equiv e^xe^y$ for real or complex numbers but not for quaternions. Yet, for the purpose of discussion, let us focus on the aforementioned case of $f$ and $g$. For pedagogical purposes:

• Do you consider $f=g$ an "identity"? What does an identity mean?
• How to convince high school students that your definition is a good one?

Answer 1

I've typically seen the term "identity" used to refer to an equation that is true wherever both sides are defined. This would include $\sin x=\tan x\cos x,$ to be sure, but it would also include $x=2x-x.$ Put another way, we say that $f=g$ is an identity iff $f$ and $g$ are identical wherever they are both defined.

Answer 2

It is customary in mathematics that when it is clear from the context that the numbers we consider are real (or complex), then the exact range for which the identity holds is left implicit. There is no difference between an identity and an equation. The claim that $f(x)=g(x)$ is, in the context where it is clear that we are talking about real (or complex) numbers, the claim that for all $x\in \mathbb R$ (or $x\in \mathbb C$) for which both $f(x)$ and $g(x)$ are defined, they are equal. Having said that, it is considered good practice (especially when dealing with students) to, as in the case you mention in the question, say something like: for all $x\in \mathbb R$ such that $\tan(x)$ is defined.

As to how to convince high school students, simply agree with them on what it means to state an identity. It's a matter of convention more than anything else, so once you agree on the convention you can check the details. In any case, this is a rather moot point to discuss as it is generally considered nit-picking and unimportant.

Answer 3

Your question is somewhat subtle. According to contemporary mathematics, two functions are identical when (1) they share the same domain and the same co-domain; (2) they act according to the same rule. Hence $\sin\theta=(\tan\theta) \cdot (\cos\theta)$ is wrong, unless you decide that $\sin$ is actually the restriction of the usual sine function to the domain of the right-hand side.

For some teachers this approach is too stiff, and they prefer to accept Cameron Buie's answer.

Edit: I approved the suggested modification, but I wrote $\sin = \tan \cdot \cos$ on purpose. Actually, it seems to me that "trigonometric identities" are not real identities between functions, but rather identities up to restricting their validity to suitable sets. But this can be done at a very elementary level: would you define $\frac{x}{x}=1$ an identity? I probably would not, since $\frac{x}{x}$ is already undefined at $x=0$ while $1$ is always defined. Somebody tends to say that this is an identity, because it is an identity on the largest set where the two sides are computable.

Answer 4

To be precise, a statement $f(x)=g(x)$ should always be accompanied by specifying the range of all free variables involved (in this case, presumably $x$ and only $x$ is free), e.g. "for all real $x$" or - in your example - "for all $\theta\in\mathbb R\setminus(2\mathbb Z+1)\frac\pi2$". On the other hand, if one works with functions, then $f=g$ automatically means: $f(x)=g(x)$ for all $x$. Even if one does not care for differences on negligible sets (e.g. sets of Lebesgue measure $0$), one should write "$f(x)=g(x)\text{ a.e.}$" to mention this; one can unambiguously use strict equality if one works with appropriate equivalence classes of functions. It is often understood that an implicit "almost everywhere" or "whenever both sides are defined" or similar should be added. Similarly, when introducing additivity of limits, one tends to memorize only $$ \lim_{n\to\infty}(a_n+b_n)=\lim_{n\to\infty}a_n+\lim_{n\to\infty}b_n,$$ but the complete statement really involves "which is to say that if the two limits on the right exist, then so does the limit on the left and its value is the sum of the two single limits" or "if two of the limits exist then so does the third and the equality holds".

So to repeat: To be careful, domain restrictions should always be explicitly stated.

Especially, an exercise definitely should be written with so much care that the domain and any exceptions are mentioned explicitly. E.g. no exercise should ask "Show that $\frac{x^2-1}{x-1}=x+1$", but "Show that $\frac{x^2-1}{x-1}=x+1$ for $x\ne1$". However, you may encounter "Hint: Use $e^{\ln x}=x$" without specification of domain (or in complex analysis: branch) if the hint is to be applied for a suitable situation (in which case your answer must include "... because $x>0$" or the like).