Timeline for How to convince my student that this is an Identity : $\sec^2x-\tan^2x=1$?
Current License: CC BY-SA 3.0
16 events
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| Mar 21, 2023 at 8:24 | comment | added | ryang | My response to this Answer is in this comment below. | |
| Jul 5, 2017 at 16:55 | comment | added | Joonas Ilmavirta | @DanielR.Collins Fair enough. It had been a while since I had read the other answers. I have never encountered a definition in my years of teaching and studying mathematics, but it might be a case of leaky memory. Personally, I prefer to keep it informal, but I know it's a matter of my taste, not of anything objective. (Also what people call "identity" in my field has an effect on this.) | |
| Jul 5, 2017 at 16:41 | comment | added | Daniel R. Collins | @JoonasIlmavirta: Michael E2's answer gave several examples. I've added a new answer with a few more (all that I find are equivalent). | |
| Jul 5, 2017 at 14:00 | comment | added | Joonas Ilmavirta | @DanielR.Collins I always considered "identity" to be an informal concept that is purposely not defined precisely. When properly formalized, they become theorems. I based my answer on what I consider the essence of an identity. At research level, "identity" in my experience means "a (meta)theorem which states that two things are miraculously equal", which is still not precise. If you know a more formal approach to defining an identity, I'm all ears (or eyes, really). | |
| Jul 5, 2017 at 13:46 | comment | added | Daniel R. Collins | -1 because this answer does not reference any definition of what an identity actually means. "Mathematical definitions mean nothing until you actually use them in anger, i.e. to solve a problem..." link | |
| Jul 5, 2017 at 7:27 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
added 117 characters in body
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| Nov 28, 2015 at 1:59 | comment | added | Ekaveera Gouribhatla | @Chan-HoSuh the problem is i could not enter into the topic "functions", because he was not been introduced relations itself. what the student knows is "an Identity is an equation which is valid for all the values(Real at present) of the variables occuring in it". Like $(a+b)^2=a^2+2ab+b^2$, $x \times 0=0$, etc. Now i need to tell him Or define separately the word "Trigonometric Identity". | |
| Nov 27, 2015 at 22:21 | comment | added | Chan-Ho Suh | @JoonasIlmavirta I get the feeling you're just talking about functions in sheep's clothing. As expressions, the two sides are certainly not equal. What you mean is their evaluations are equal for certain inputs, and so functions are lurking there whether one spells it out or not. | |
| Nov 27, 2015 at 21:10 | comment | added | Chan-Ho Suh | Benoit, yes that should be "constant function" not "identity function". However many people in the US write functions with the variable in the expression, eg f(x) rather than f. This is particularly commonplace in precollege mathematics. Even professional mathematicians will often write d/dx sin x = cos x when they are asserting functional identity (trig functions tend to have their own special notational conventions). | |
| Nov 27, 2015 at 20:47 | comment | added | Benoît Kloeckner | @Chan-HoSuh: there is an equality of functions, but it would rather be written $\sec^2-\tan^2=1$ (and then your remark about domain of $1$ applies; note that here $1$ would be called a constant function rather than an identity function). | |
| Nov 27, 2015 at 20:29 | comment | added | Joonas Ilmavirta | @Chan-HoSuh, I would not regard this as an equality of functions. Instead, I see it as an equality of expressions. The two expressions are equal for certain values of the variable. You can of course view it as an equality of functions, but I feel that adds a layer of unnecessary structure. This is a matter of taste, but it is certainly not a must to relate it to functions. | |
| Nov 27, 2015 at 20:00 | comment | added | corsiKa | In fact, failure to put the domain of an identity in is usually the root cause of these silly "1 = 0 proofs" we see high schoolers engage in. | |
| Nov 27, 2015 at 18:40 | comment | added | Chan-Ho Suh | This answer is halfway there, but it obscures what might be a subtlety to most students. An identity like the above is really a statement of equality between functions. Although the right hand side looks like a number rather than a function, "1" really is referring to an identity function and thus you also need to clarify the domain for it. | |
| Nov 27, 2015 at 14:06 | comment | added | Ekaveera Gouribhatla | Thanks that was pretty clear | |
| Nov 27, 2015 at 14:06 | vote | accept | Ekaveera Gouribhatla | ||
| Nov 27, 2015 at 13:24 | history | answered | Joonas Ilmavirta | CC BY-SA 3.0 |