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I have some problem understanding the identity in math . I found an identity $$\tan x=\frac1{\cot x}$$ How come it can be an identity when $\tan0$ exists but $\frac1{\cot0}$ does not because $\cot0$ is undefined. If Above identity is true, please explain what to use when we have value of $\tan0$?

AnDromeda
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    We can say that $\tan{x} = \frac{1}{\cot{x}}$ whenever $\cot{x}$ and $\tan{x}$ are defined. – Minus One-Twelfth Oct 28 '21 at 10:53
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    @MinusOne-Twelfth Whenever both $\tan x$ and $\cot x$ are defined, to be precise. – Milten Oct 28 '21 at 10:53
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    That being said, for $x\to 0$ we have $\cot x\to\pm \infty$, so it's not too far of a stretch to say the identity holds even for $x=0$. With a grain of salt. – Milten Oct 28 '21 at 10:56
  • @Milten Then we can not use this identity when x=0 ? right ? I mean if we have a situation where tan(0) is included – AnDromeda Oct 28 '21 at 10:57
  • @AnDromeda In a rigorous sense, no. – Milten Oct 28 '21 at 10:57
  • @Milten In many math examples they include tan(0) value but our teachers keep using this identity – AnDromeda Oct 28 '21 at 11:00
  • @AnDromeda It's a question of context and even personal taste (I know, that actually exists in maths too!). To say anything more, it would help if you could one of those "many math examples" concretely. – Milten Oct 28 '21 at 11:03
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    It's an identity yes, but really it's a definition. What does the cotangent function mean? Its geometric definition shows that when $\tan(x)\neq0$, $\cot(x)=\frac{1}{\tan(x)}$, and when $\tan(x)=0$ the geometric definition gives $\pm\infty$ as the most sensible answer anyway. Alternatively, the function $\tan(x)\cdot\cot(x)$ is everywhere equal to $1$ - the singularities become removable - so the identity does make sense. Which examples that came up in class are confusing for you? – FShrike Oct 28 '21 at 11:04
  • @FShrike it mostly occurs in proofs where we need to prove one side to other . In such cases it is not asked about value and in some kind of intervals where value at x=0 is included . So based on your answers it will not be wrong , right ? I can use this identity , or it is better to show that it is true when x!=0? – AnDromeda Oct 28 '21 at 11:12
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    Well it is definitely possible to make careless and unrigorous mistakes, especially during integration, if you fiddle with $\tan$ and $\cot$ at "undefined" values, so you'll need to provide a specific example. Moral of the story is, it is a definition, and if you are working over an interval where singularities are involved then you need to be careful. A common theme in proofs is the idea of "recovering cases"; sometimes, for the sake of a step in a proof, you assume $x\neq0$, for example, but when you're done you have an expression that works for all $x$: the case $x=0$ has been recovered – FShrike Oct 28 '21 at 11:14
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    @FShrike Thanks brother for the answer – AnDromeda Oct 28 '21 at 11:21
  • @FShrike Can I answer to my question like that : Identity is not true when one of the sides of the equation is not defined , it depends on the domain of the variable ? – AnDromeda Oct 28 '21 at 11:30
  • Yes, an identity is an equation that is universally true on the domain of discourse, i.e., wherever the variable tuple is defined. (I say tuple in case there are multiple variables, e.g., in $(x+y)(x-y)=x^2-y^2$.) – ryang Nov 24 '21 at 10:42

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