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What is the correct meaning of term "formula"?

My teacher says that a formula is valid for all values of variables we plug in. Is it true that we can say formula is valid for all defined values of variable we plug in? I got this doubt when he told me the "formula"

$$(\tan(x))^2(\sin(x))^2 = (\tan(x))^2 - (\sin(x))^2\,.$$

Is it correct to say it a formula? Or is this just two functions equal to each other for the values of $x$ for which they are defined?

Mike Pierce
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Lalit Tolani
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    As far as I'm aware, you're describing an "identity": two different looking expressions that are actually identical for all inputs. A "formula" is a recipe for computing some desired quantity, such as the formula for the area of a circle or the quadratic formula. – Austin Mohr May 21 '21 at 01:28
  • My answer is here. – ryang Feb 07 '22 at 11:15

2 Answers2

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The terms "formula" and "equation" can be thought of as synonyms, but the word "formula" has a specific connotation to it. I think of a formula as more of a recipe for a specific output. The formula for distance is the equation $d = rt$, which is a recipe for calculating the distance something travels $d$ given its rate of travel $r$ and duration of travel $t$ as ingredients.

I wouldn't describe what the equation you wrote as a "formula".

Mike Pierce
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  • sir how could you describe the equation i write, simply an equation or something like identity – Lalit Tolani May 21 '21 at 02:04
  • @lalittolani Yeah I like the word "identity" for it since that equation is manifestly true for any argument $x$. That's as opposed to an equation like $y = 3x+4$ that is either only true for certain pairs $(x,y)$, or that you must declare to be true and intrinsically restrict your attention to those same $(x,y)$ anyways. – Mike Pierce May 21 '21 at 03:20
  • can an identity have a set of values of variable such that it is defined for values of variable which are part of that set only – Lalit Tolani May 21 '21 at 03:49
  • @lalittolani I mean, yeah? Your equation is only an "identity" on the $x$ that are in the domain of $\tan$ and $\sin$. But to call something an "identity" you probably want it to hold true on the entire maximum (implied) domain of inputs like this. It would be silly to say something "the equation $x^2=9$ is an identity for $x$ in the set ${-3,3}$. Not much of an "identity" that would be. – Mike Pierce May 21 '21 at 04:47
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A formula can be seen as an expression that returns a value given $n$ number of parameters. An example is $$A=S^2$$ This "formula" gives the area of a square with side length $S$. On the other hand, a function is a special type of relationship between two sets called domain and codomain: each element in the domain is mapped to one and only one element in the codomain. Now, the example you were given is usually called a trigonometric identity, you can look at it as equality between two formulas, it is not a formula itself.

DouL
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