I taught three basic trigonometric identities. One of my 9th grade students asked: How can we say $$\sec^2x-\tan^2x=1$$ is an identity since when we plug in $x=\frac{\pi}{2}$ the identity fails which creates an ambiguity for the definition of an identity in mathematics?
How should I try to explain him without entering into calculus part?
An identity always holds for some values of the free variables. Sometimes the allowed values are all real numbers, sometimes something else. In this case the identity holds whenever $\sec x$ and $\tan x$ are defined (they are defined in the same set). In a certain limiting sense the identity is also true at $x=\pi/2$, but you probably don't want to go into that with your student. The point is: An identity should always come with information about when it is supposed to hold. It's not just an equation.
Even with the more common identity $\sin^2x+\cos^2x=1$ you can start asking questions: This holds for geometrical reasons for $x\in(0,\pi/2)$, but does it hold for all $x\in\mathbb R$? Ok, does it hold for all $x\in\mathbb C$ as well? Well, what if $x$ is a matrix (you can still define the functions via the matrix exponential)? The answers are all yes, but certainly only the real case is meant when it is first encountered. It is not clear how far an identity should extend unless you explicitly tell what you mean by it. You cannot always trust that your students will intuitively see what values the variables should take.
For comparison, the Pythagorean theorem does not state just $a^2+b^2=c^2$. It states this equation and tells what the three variables should be for this to hold. The equation only becomes meaningful once there is a triangle with a right angle and the lengths of the sides are named appropriately — it is all this information together that makes the theorem.
If $x$ is meant to be a real variable, then your student is right; $\sec^2 x - \tan^2 x = 1$ is not* an identity; it suffers from the same sort of problems that $\frac{x}{x} = 1$ does.
It fails to be an identity in two different ways, depending on fine details of what one actually means: either the left hand side is undefined because $x$ is not restricted to the domain of $\sec$ and $\tan$, or the identity fails because the domain of the left hand side differs from the domain of the right hand side.
(the former is syntax/semantics based on total functions, the latter on partial functions)
To make a correct statement, limit $x$ appropriately: e.g. $\sec^2 x - \tan^2 x = 1$ whenever $x \neq (2n+1) \frac{\pi}{2}$.
As another answer suggested, the version $1 + \tan^2 x = \sec^2 x$ has the advantage of actually being an identity in the partial function syntax/semantics when $x$ is a real variable, since both sides have the same domain. Think of it as saying "if one side is defined, then so is the other and they're both equal".
*: Technically, there are other settings where this would be an identity; e.g. if the two sides don't refer to functions, but instead to equivalence classes of functions where two functions are equivalent if they differ on a set of measure zero, or if every operation is to be followed by "... then continuously extend the result"
One answer is to say that, properly, the identity is $\sec^2x=1+\tan^2x$, where the sides fail to be defined at the same values.
For typical inputs the equality holds. Thus, whatever further qualifications or discussion may be interesting or necessary, we're "most of the way there". To deny that it's "an identity" is to pretend to disqualify the fact on some technical grounds that do not address significant issue.
This question simply turns on the definition of the word "identity". Michael E2's answer is currently the best one, because he actually references a definition. Any college algebra or precalculus book that I can touch has such a definition, and every one that I see is effectively equivalent. In addition to Michael E2's examples, I can add:
Ratti & McWatters, Precalculus: A right triangle approach, 2E (Sec 1.1) [2011]:
An equation that is satisfied by every real number in the domain of the variable is called an identity.
Sullivan, Algebra & Trigonometry, 7E (Sec 1.1) [1987-2005]:
An equation that is satisfied for every value of the variable for which both sides are defined is called an identity.
Rietz & Crathorne, Introductory College Algebra, Revised (Ch. II, Art. 18) [1923-1933]:
The two members of an identity are equal for all values of the symbols for which the expressions are defined.
I include dates and the latter example to highlight that the customary definition of an "identity" has been stable and consistent for at least the last century.
The OP should refer the student to the definition in use in the course's textbook and/or lecture notes (which hopefully matches all of the preceding examples). In particular, an identity only needs values to match when both sides are defined; so it is not foiled by looking at values outside the domain of the expressions. This answers the student's question.
For a similar example that one should be preparing students to look very carefully at their definitions, consider the fact that the definition of a continuous function similarly looks only at points in the domain (so that, e.g., $1/x$ qualifies as a continuous function).
$π/2$ and $−π/2$ are not in the domain. I'd take a step back and offer examples from algebra where, when observing an equation, the first discussion is "what is the domain?"
Start with $\sin^2(x) + \cos^2(x) = 1$, which is defined for $0\leq x\leq 2\pi$.
Then divide through by $\cos^2(x)$ to get $\tan^2(x) + 1 = \sec^2(x)$, which is not defined when either of the denominators $= 0$.
Rearrange to get $\sec^2(x) - \tan^2(x) = 1$, as per the question.
The reason it is an identity is because it derives from an identity but is undefined when trying to divide by zero.
Also, I always taught an identity with an = sign with 3 bars ($\equiv$). This may not be common in other parts of the world, as suggested by some of the answers above, which are a bit stuck on the meaning and purpose of the 'ordinary' = sign: I suggest don't use it.
For a right triangle with angle $x$, hypotenuse $H$, adjacent side $A$ and opposite side $O$:
$sec(x) = H/A$
$tan(x) = O/A$
$sec^2(x) - tan^2(x) = 1$
$H^2/A^2 - O^2/A^2 = 1$
$H^2 = A^2 + O^2$, which is just the Pythagorean Theorem
Look at the "identity". On the right-hand-side, you apparently have the number 1 and on the left hand side, you have the difference of two functions (which is also a function). A number is not a function, so the stated identity would seem to be wrong. Undoubtedly the student is trying to make sense of this by thinking of "=" as some sort of process where it really means (or so s/he thinks) "plug in numbers on the left-hand-side and the result should be the same as the numerical value on the right". This is wrong.
The identity is in fact asserting equality of the two sides. You just have to be careful about defining what the sides are. The right hand side, which is represented by the symbol "1", is really the constant function giving 1 on a specified domain. It would be better to represent it as something like $1_D$ where $D$ is defined to be the domain that matches the domain of the left-hand-side.
