Talk:List of trigonometric identities

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These identities are special because this one is about constructing regular polygons and the minimal polynomial for 2cos(2π/n).

${\displaystyle 6\cos \left({\frac {2\pi }{7}}\right)=2{\sqrt {7}}\cos \left({\frac {1}{3}}\arccos \left({\frac {\sqrt {7}}{14}}\right)\right)-1}$

${\displaystyle 12\cos \left({\frac {2\pi }{13}}\right)={\sqrt {104-8{\sqrt {13}}}}\cos \left({\frac {1}{3}}\arctan \left({\frac {{\sqrt {3}}({\sqrt {13}}-1)}{7-{\sqrt {13}}}}\right)\right)+{\sqrt {13}}-1}$

${\displaystyle P(x)=\sum _{n=0}^{k}(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }{\binom {k-\left\lfloor {\frac {n+1}{2}}\right\rfloor }{\left\lfloor {\frac {n}{2}}\right\rfloor }}x^{k-n}}$

${\displaystyle \Rightarrow P\left(2\cos \left({\frac {2\pi }{2k+1}}\right)\right)=0}$

Does anyone agree that they should be added? 173 Ascension 257 (talk) 18:31, 28 December 2022 (UTC)

What's the source? Apocheir (talk) 01:39, 29 December 2022 (UTC)

The source of the first two identities is made by American mathematician Andrew M. Gleason, who has published these identities, from:

http://math.fau.edu/yiu/PSRM2015/yiu/New%20Folder%20(4)/Downloaded%20Papers/AMMGleason1988.pdf

The polynomial was something else, which is from:

https://en.wikipedia.org/w/index.php?title=Minimal_polynomial_of_2cos(2pi/n)&oldid=1070293024#cite_ref-3

The sequence that I found is from:

https://oeis.org/A187360/a187360.pdf

...and the cases for n = 1 to 30 seem to match what the series gives.

I could hardly find the exact polynomial from any other source, yet these sources are related to the series that gives the minimal polynomial of ${\displaystyle 2\cos \left({\frac {2\pi }{2k+1}}\right)}$, or ${\displaystyle 2\cos \left({\frac {2\pi }{n}}\right)}$ for all positive odd numbers n, yet these related articles may help:

https://www.jstor.org/stable/2324301?read-now=1&seq=1#page_scan_tab_contents

https://www.jstor.org/stable/2301023?read-now=1&seq=1#page_scan_tab_contents 173 Ascension 257 (talk) 23:12, 30 December 2022 (UTC)

Summarize Timeline Fact Check

Discussion is opened as per recommendation of user @Apocheir. Let us give some time for this to be discussed here before choosing to remove content. I have added the section with citation so there is no harm done in the time being. EditingPencil (talk) 21:09, 30 November 2023 (UTC)

Mostly my issue was seeing yet another uncited identity (albeit a familiar one) added to the page. That said, I question whether either of these are useful as trigonometric identities. As derivations of the relationship between trig functions and the exponential function, or as approximations of the functions, sure, but that's not what this page is about.

The Taylor expansion of sine and cosine is covered adequately on Sine and cosine. The product formulas, well, maybe I'm not well enough versed in special functions to immediately see what their point are, but they don't strike me as very useful. This page is so long that Who Wrote That? chokes before it can tell me who added it and when.

The goal of this page is not to repeat every identity in Abramowitz and Stegun. (That might be a good WikiSource or WikiBook project, though.) Apocheir (talk) 21:59, 30 November 2023 (UTC)

Fair but I disagree. It takes up very little space and it is very useful when you have to express trig functions in the n-th order and those formula can be used to prove things as well. In fact, those two trig functions can alternatively be defined as such to begin with, like otoh, Andrew Gleason does in his book on Abstract analysis. I didn't initially bother citing them because of how often I come across them as a student in physics. The product formula is less useful in comparison, although my knowledge of this is also very limited. EditingPencil (talk) 23:20, 30 November 2023 (UTC)

@Apocheir Let me know if this resolves the issue. Otherwise, I recommend we give time for others to comment before making changes. Those 5 extra lines are not really an urgent matter and they're even helpful imo.

PS: I reverted one of your edits due to wrong use of terminology, I also recommend you discuss that here before trying to merge the sections. Thanks for understanding! ^^ EditingPencil (talk) 23:30, 30 November 2023 (UTC)

Making the change first, and then finding out if anyone disagrees with it, is the standard practice on Wikipedia. Read WP:BRD. Apocheir (talk) 23:52, 30 November 2023 (UTC)

I meant you can discuss if you are planning to undo reversion. EditingPencil (talk) 00:09, 1 December 2023 (UTC)

I think we should add an article called trigonometric identity which consists predominantly of prose (rather than formulas) and discusses the most important identities along with general material about the context (why were identities historically important, why do they appear more in trigonometry than other subjects, how are they used in education, etc.), and then we can leave this article as a sink to park arbitrarily much stuff that people can find sources for. Anything especially important should also have its own dedicated article.

In general "list" articles aren't ever very legible, and I don't think this one can really be put into any great kind of organization. If it ends up reproducing every relevant bit of material in Abramowitz and Stegun, that seems fine. That's still useful as a reference. –jacobolus (t) 00:47, 1 December 2023 (UTC)

• I think we should remove these infinite sums and products for the reason that they are not really trigomentric identities, which should i.m.o. have at least two trigonometric functions in the equation. - DVdm (talk) 23:56, 30 November 2023 (UTC)

  Then that is also what the article must state at the beginning. Instead, these equations are fine under that description. I think removing these is unnecessary. I think the discussion should also include whether this section can be pushed off into 'miscellaneous'. EditingPencil (talk) 00:14, 1 December 2023 (UTC)

  btw proofs on trigonometric identities has few extra identities which have only one trig function but I wont edit them here since they can be trivially derived from infinite series expansion. EditingPencil (talk) 00:23, 1 December 2023 (UTC)

  Proofs of trigonometric identities is a terrible scope for an article. There's no tight unifying theme, there's no obvious narrative organization, and the possible scope is almost unbounded. The current version is also entirely unsourced, and some may be original research.

  That article should be split apart with proofs moved to dedicated articles about each separate identity if they are relevant. Some of it can just be deleted; not every math proof is particularly encyclopedic. Some instead belongs in a textbook or the solutions manual of a problem workbook. –jacobolus (t) 00:53, 1 December 2023 (UTC)

Summarize Timeline Top Qs Fact Check

I added the following half-angle formulas: $${\displaystyle {\begin{aligned}\sin {\frac {\theta }{2}}&={\frac {\sin \theta }{\sqrt {2(1+\cos \theta )}}},\quad |\theta |<\pi \\[3pt]\cos {\frac {\theta }{2}}&={\frac {\sin \theta }{\sqrt {2(1-\cos \theta )}}},\quad 0<\theta <2\pi \\[3pt]\sec {\frac {\theta }{2}}&={\frac {\sqrt {2(1-\cos \theta )}}{\sin \theta }},\quad 0<\theta <2\pi \\[3pt]\csc {\frac {\theta }{2}}&={\frac {\sqrt {2(1+\cos \theta )}}{\sin \theta }},\quad |\theta |<\pi \\[3pt]\end{aligned}}}$$ Are they correct? Eric Kvaalen (talk) 12:19, 30 September 2024 (UTC)

I reverted your edit per policy WP:PROVEIT. It doesn't matter whether they are correct. You need sources for them, if only to demonstrate notability. This is correct too:

${\displaystyle \sin {\frac {\theta }{2}}+\sin {\frac {\theta }{2}}=2\sin {\frac {\theta }{2}},}$

but we don't add it to the aticle because it's only notable when a relevant reliable source mentions it. This was amply explained to you before:- see User_talk:Eric_Kvaalen#Original_research - DVdm (talk) 13:46, 30 September 2024 (UTC)

Ah, yes, you're the one who wouldn't let me put how much the planets pull on the earth. And you said something on my talk page about how you spend your time preventing people looking for interesting things on Wikipedia. I didn't understand that. So do we need a reference for everything that is said in Wikipedia? Even something like ${\displaystyle {\frac {\sin \theta }{{\sqrt {2(1+\cos \theta )}}{\sqrt {(1-\cos \theta )/2}}}}=1}$ when ${\displaystyle \sin \theta >0?}$ Eric Kvaalen (talk) 11:09, 1 October 2024 (UTC)

No, I said something on your talk page about how I you spend some of my time preventing people finding uninteresting things on Wikipedia.

Yes, even, and specially something like that. See wp:BURDEN. If you don't like it, there are other places. - DVdm (talk) 11:55, 1 October 2024 (UTC)

Well, you wrote, "I spend some time preventing others to spend a lot of time looking for things that people have put into Wikipedia because they are interesting..." Whatever. Anyway, why do you spend your time enforcing your idea of what is interesting? And what is the point of requiring references for things like this? Eric Kvaalen (talk) 15:08, 1 October 2024 (UTC)

Just read the policies. I spend some of my time guarding them. I will repeat in bold: the point of requiring references for things like this, is to demonstrate notability. If you indeed find this impossible to understand, then I can't help you. - DVdm (talk) 15:15, 1 October 2024 (UTC)

@Eric Kvaalen: The point of this page is not to exhaustively list every possible trigonometric identity that any Wikipedian can make up, which would be overwhelmingly large and not very useful to readers, but rather to organize a list of the most important and influential trigonometric identities, as identified by the best secondary sources we can find. Unfortunately this page doesn't currently live up to its ideal, and there are identities missing which should be listed, identities mentioned which should be elaborated, pictured, or even turned into dedicated articles, and also identities listed which should be cut or removed to more specific articles. The current content of this page is mostly (all?) supportable by reliable sources but not all of it is currently well sourced – that should also be improved. But if you want to add new identities that nobody has yet thought to include in the past two decades since this page was created, you need to provide a source demonstrating that the identity is considered important. Even with sources, we can have discussion here about whether each identity is really pulling its weight, and might decide to trim some material for length. If you discover a new identity that is not included in reliable secondary sources, the appropriate place to publish it is in a peer-reviewed paper (or a preprint, or a blog post, etc.), not here. –jacobolus (t) 15:45, 1 October 2024 (UTC)

Indeed, that's why I also undid this and this. - DVdm (talk) 16:02, 1 October 2024 (UTC)

@Jacobolus: I'm not trying to list every possible identity! The one in the article for ${\displaystyle \sin \theta /2}$ is inaccurate on a computer when θ is small, because you have subtraction of two similarly sized numbers (1 and ${\displaystyle \cos \theta }$). The same problem occurs for the others (cos, sec, and csc), either for small θ or for θ near π. This is all very basic computer science and I'm sure it's in the literature, since it's important. Instead of destructively deleting, editors should help by finding references, if they really think it's necessary to prove that the alternative formulas are better in certain cases, even though it's fairly obvious. Eric Kvaalen (talk) 19:38, 1 October 2024 (UTC)

If it is in the literature and is important, then you should find a source, which shouldn't be hard. –jacobolus (t) 19:44, 1 October 2024 (UTC)

Note that ⁠${\displaystyle {\tfrac {1}{2}}(1-\cos \theta )}$⁠ is the haversine. There are several ways you can compute it accurately if you need a formula that works for small angles; what kind of formula would be useful would depend on the context. (The sine half-angle formula you listed here is a fine enough choice.) The cosine, secant, and cosecant of a half angle don't have much use IMO. –jacobolus (t) 20:22, 1 October 2024 (UTC)

@Jacobolus: Well, the best way to calculate haversine of a small θ is ${\displaystyle (\sin(\theta /2))^{2}}$. But our problems is to calculate ${\displaystyle \sin(\theta /2),}$ so that doesn't help. And we want to find it from trigonometric functions of θ, not from θ itself. In other words, in applications we often have the sine and cosine of θ, and we want ${\displaystyle \sin(\theta /2).}$ We don't want to find θ first and then simply divide by 2 and take the sine. Eric Kvaalen (talk) 19:53, 2 October 2024 (UTC)

@Jacobolus and Eric Kvaalen: it's best to go on about these technicalities on one of your user talk pages, as, per wp:talk page guidelines, we cannot discuss them here in article talk space. We can have discussions about the existence (or quality) of sources, but, unless referring to sources, not about best ways of calculating anything - DVdm (talk) 08:42, 3 October 2024 (UTC)

@DVdm This seems like an unhelpfully hostile response to a quite ordinary Wikipedia talk page discussion. The need for sources has already been mentioned multiple times. –jacobolus (t) 13:20, 3 October 2024 (UTC)

Wasn't meant to be hostile, sorry about that. Just as a reminder of the guidelines. Cheers. - DVdm (talk) 15:50, 3 October 2024 (UTC)

(No worries, and maybe "hostile" is stronger descriptor than I quite meant. But "who would use this identity in practice in what context" seems like an entirely reasonable talk page question to help decide what belongs on a page like this. I agree with you anything added needs a source, and I expect Eric Kvaalen can find one if they go search for it, at least for the half-angle sine identity.) –jacobolus (t) 16:43, 3 October 2024 (UTC)

I think a section on identities used to improve numerical accuracy near certain values would be a good addition, if referenced to the literature.

Doesn't the half-angle cos identity that was removed have the same problem as the half-angle sin identity in the article? Maybe even worse since it's dividing a small quantity by a small quantity near 0 and π... Apocheir (talk) 22:09, 3 October 2024 (UTC)

There's not much issue for cosine because cosine of a small angle approaches 1, so numerically you eventually just bump into 1 and run out of significant digits; the relative error inevitably goes to 0 as your angle gets smaller. (The formula given at the top of this section approaches 0/0, but I haven't tried to figure out how it behaves numerically in approximate arithmetic.) There's a much bigger issue for the versine (⁠${\displaystyle \theta \mapsto 1-\cos \theta }$⁠), sine, tangent, chord, etc. –jacobolus (t) 22:26, 3 October 2024 (UTC)

@Apocheir and Jacobolus: Yeah, there is a problem with "my" cos(θ/2) formula around θ=0 (even though it goes toward 0/0, because the denominator is inaccurate), but it avoids that problem around θ=π, whereas the formula in our article has the problem there. That's why both formulae should be given. Why do we need references for things like this which are easily worked out? This is math, not history! Eric Kvaalen (talk) 10:25, 4 October 2024 (UTC)

Because it is way beyond wp:CALC. We do need references for things like this, to show that they are sufficiently interesting for an encyclopedia. That is by design. - DVdm (talk) 11:28, 4 October 2024 (UTC)

The point of Wikipedia is to describe and summarize the content of existing literature, not to present novel results. See Wikipedia:No original research. –jacobolus (t) 14:06, 4 October 2024 (UTC)

@DVdm and Jacobolus: I looked at the link that DVdm provided, and this is what I found: "Wikipedia has no firm rules"! Aren't we able to decide whether something is interesting? I think it's insulting to us Wikipedia editors to say that we can't decide for ourselves whether something is interesting. I had a big argument a while ago when I pointed out that the dates of Easter repeat after a certain time. The others said I had to prove that that is interesting by finding a reference! I think that's ridiculous. And it doesn't take "research" to show that the equations I added here are not only correct but more accurate in certain areas. It's insulting to say that Wikipedia editors are not capable of such simple mathematics and they have to find it written in some book! I only have two math books, one on engineering mathematics, and one on abstract algebra. Ah, plus another on a rather arcane subject. So none of my books will have these simple formulas. Eric Kvaalen (talk) 21:20, 5 October 2024 (UTC)

@Eric Kvaalen "Wikipedia has no firm rules" doesn't mean the same as "intentionally do the opposite of several longstanding Wikipedia policies whenever you personally feel like it, for no particular reason". If you want to push on this one, take it up at e.g. Wikipedia:Village Pump (policy) or Wikipedia talk:No original research.

But the most important and fundamental of all Wikipedia rules is that Wikipedia content is decided by consensus, both on individual pages and site-wide. You certainly haven't demonstrated any local consensus here for adding Wikipedians' new unpublished research into trigonometric identities. –jacobolus (t) 21:29, 5 October 2024 (UTC)

Ditto. - DVdm (talk) 21:47, 5 October 2024 (UTC)

@DVdm and Jacobolus: What I don't understand is what motivates you to enforce the rules (as you interpret them). Nobody requires you to do that. Do you really think the formulas I added are incorrect? No. Do you really think they are useless? No. Do you really not understand that they are better in certain cases? No, you do understand that. Do you really think that readers will find those formulas extremely boring and will think the article is worse if they are there? No, I don't believe you think that. Eric Kvaalen (talk) 12:22, 6 October 2024 (UTC)

Even with a source I would be hesitant to add this formula to this article, because while I can trivially derive it myself, having now done about 20 minutes of literature search I couldn't find it anywhere, which indicates that it's really not widely known/used, and probably doesn't belong on a generic page like this. With a source, I think it would be more appropriate to include somewhere in a dedicated article half-angle formulas (currently a redirect). I think this topic should be expanded into a separate article either way, because List of trigonometric identities does not have the space for description of the context, proofs, images, history, applications, etc.

To answer your other questions: I do really think 3/4 of these formulas are pretty useless and the fourth one has an extremely niche use which I'm having trouble imagining a clear concrete situation for, but I have myself used similar (novel, niche) formulas of this type in the past, in various one-off situations, so I'm sure someone, somewhere could come up with a use for it. That's not good enough for inclusion into Wikipedia though. I can personally think of / have myself derived several dozen more trigonometric identities which are not included on this page which I think are more useful than this one, some of which can even be found in (obscure) sources. If I had been carefully keeping a list of every moderately interesting trigonometric identity I ever saw in a published paper or book somewhere, that would be many hundreds more formulas, again mostly more useful/interesting than these. But adding them here would not really be net helpful; it balloons the page for something that isn't really relevant to readers, which has the effect of hiding what they might be looking for in the noise.

I think you misunderstand Wikipedia's nature. Wikipedia is an encyclopedia, a tertiary source, not a primary source. Its purpose is to organize what reliable secondary sources have already reported/discussed, not to be a venue for new research results. If you publish your formula somewhere, and it proves to be influential, then it might naturally wind up in Wikipedia a few years down the line.

Please read Wikipedia:What Wikipedia is not § Wikipedia is not a publisher of original thought and Wikipedia:No original research. Again though, this isn't really the venue for arguing bedrock Wikipedia policies which have been the solid consensus of editors for 20 years. –jacobolus (t) 15:27, 6 October 2024 (UTC)

@Eric Kvaalen: what motivates me to do anything is Off-topic on an article talk page. You can ask me on my user talk page or on yours. - DVdm (talk) 16:20, 6 October 2024 (UTC)

Well, DVdm, it's not to me that you need to give account for your deeds. I am not Jesus Christ. II Cor 5:10

@Jacobolus: I don't think an encyclopedia has to be a "tertiary source". Diderot said the Encyclopédie was "to change the way people think" and for people to be able to inform themselves and to know things (Encyclopedia). I strongly disagree with the idea that we editors cannot use our brains enough to make obviously true inferences or to point out things, without having a reference. A while back I wanted the mass of nitrogen in the earth's atmosphere, and I went to Atmosphere of Earth to find out how much nitrogen there is in the air, in mass percent. The number wasn't there, so I did a quick calculation of the mass-based composition of air, from the molar composition, and added it to the article. But a certain guy (with whom I had had arguments previously) reverted it, saying I didn't have a reference! Look, I have tried to argue this in other venues. Please read Wikipedia:Village pump (policy)/Archive 152#Are policies being used to the detriment of Wikipedia? and then Wikipedia:Village pump (policy)/Archive 156#Are policies being used to the detriment of Wikipedia?. That was a discussion in 2020 but it was closed down by an administrator who didn't agree with me, even though there were more people who did agree with me.

I think we should have the right (and I think we do have) to make this article better by adding explanation of why certain formulae should be used rather than others. If you think there are too many formulae, you should be allowed to use your head and decide what should and shouldn't be here. The idea that "Wikipedia is not a publisher of original thought" should not be used to prevent editors from thinking at all! It's amazing what gets called "original research"! Such as calculating the mass-percentages of gases in air.

I'm curious what kinds of trigonometric identities you have come up with, and why!

Eric Kvaalen (talk) 08:53, 9 October 2024 (UTC)

Off-topic. See wp:talk page guidelines. Warning for abuse on your user talk page. - DVdm (talk) 09:05, 9 October 2024 (UTC)

"I strongly disagree with the idea ..." – That's fine, but Wikipedia runs on consensus, not what one editor personally prefers. You should take this up elsewhere because it is off topic here, and Wikipedia talk pages are not a general-purpose discussion forum. Please read ... – Okay, I looked at these. It appears that you have a longstanding habit of adding unsourced or poorly sourced claims to Wikipedia and then starting long protest discussions whenever they are reverted. I would urge you to desist from such behavior, which seems like trolling for an argument. It's disruptive to the Wikipedia project, and if you keep at it long enough someone will eventually get annoyed enough to start formal processes demanding that you stop. –jacobolus (t) 13:44, 9 October 2024 (UTC)

Jacobolus, that's quite an unfair conclusion to draw from that discussion! Eric Kvaalen (talk) 06:57, 10 October 2024 (UTC)

Summarize Timeline Top Qs Fact Check

The whole section needs to be revised. First of all, the source quoted points to the homepage of MathWorld, instead of the correct page: https://mathworld.wolfram.com/Multiple-AngleFormulas.html. Second, the formulas for sin(nx) and cos(nx) on the Wiki page are different from the ones written in the source, and the source doesn't even contain a formula for tan(nx), just a few examples. Third, I tested the formula on the Wiki page for cos(nx), and it produces the right results for n=2 and n=3, but the wrong one for n=4. Took me a few days to understand why my calculations were correct and my results inexplicably wrong. :( 194.230.144.177 (talk) 14:42, 2 November 2024 (UTC)

I added both sums.

I did not get them from the MathWorld Webpage. I derived them myself. A lot of the proof was merely inferred out of the patterns of the cosines. I added them anyway due to their relevance. Absnone (talk) 04:05, 13 January 2025 (UTC)

For the term "sgn(a)=sqrt(a^2+b^2)," is "sgn" a typo of "sin"? Also, for the equation in "Arbitrary phase shift," is the second term intended to be cosine, not sine? Finally, if we take the first derivative of this equation, we see that a linear combination of sine and cosine equals something in terms of cosine. Is it worth including that equation as well? Physics170 (talk) 18:58, 7 April 2025 (UTC)

is "sgn" a typo of "sin"? – No. See Sign function.

is the second term intended to be cosine – Also no. You could make a variant of these with cosines instead of sines, but I don't think it would help readers. –jacobolus (t) 20:39, 7 April 2025 (UTC)

Summarize Timeline Top Qs Fact Check

The notation converting the product (big pi) into sum (big sigma) is unclear. The index of the summation is given as e, but then e is subscripted in the argument of the summation. In one iteration of the sum, if all of the e values are the same, then it should just be e, not a subscripted e. If, in one iteration of the summation, all of the angle values are the same, then they should not have different subscripts. If it is difficult to express this because of the triple dots in the middle, perhaps the argument of the cosine inside the summation should be expressed using a summation, and each summation can have a clearly labeled index. — Preceding unsigned comment added by Pauldebarros (talk • contribs) 18:37, 6 June 2025 (UTC)

The version there now is trying to say that you should take all possible sums of the several angles with either sign. For example with three angles, you would have: $${\displaystyle {\begin{aligned}\cos \theta _{1}\cos \theta _{2}\cos \theta _{3}&={\tfrac {1}{8}}{\bigl (}\cos(\theta _{1}+\theta _{2}+\theta _{3})+\cos(\theta _{1}+\theta _{2}-\theta _{3})\\&\qquad \quad {}+\cos(\theta _{1}-\theta _{2}+\theta _{3})+\cos(\theta _{1}-\theta _{2}-\theta _{3})\\&\qquad \quad {}+\cos(-\theta _{1}+\theta _{2}+\theta _{3})+\cos(-\theta _{1}+\theta _{2}-\theta _{3})\\&\qquad \quad {}+\cos(-\theta _{1}-\theta _{2}+\theta _{3})+\cos(-\theta _{1}-\theta _{2}-\theta _{3}){\bigr )}.\end{aligned}}}$$ Maybe you can make some proposed improved notation more explicit by pasting proposed notation in this discussion. I'm not quite sure what you have in mind. –jacobolus (t) 19:11, 6 June 2025 (UTC)

Summarize Timeline Fact Check

I would like to ask for a policy-based opinion regarding the inclusion of trigonometric identities.

These identities are currently being discussed on Russian and Ukrainian Wikipedia, but neither community is active enough to reach a clear consensus.

• Russian Wikipedia discussion: ru:Обсуждение:Тригонометрические тождества § Добавить тождество
• Ukrainian Wikipedia discussion: uk:Обговорення:Список тригонометричних тотожностей § Геометричне доведення значення and § Обговорення тотожності

The question is not whether the identities are mathematically correct, but whether those inclusion in a list article would be acceptable under Wikipedia policies, especially: WP:V (verifiability), WP:NOR (no original research), and WP:RS (reliable sources)

The identities appear in the literature in [textbooks / academic sources], but is not always listed explicitly in standard "list of identities" articles.

Any guidance would be appreciated. Ambitious miss (talk) 05:25, 13 December 2025 (UTC)

Do you have a source? (A source in English would be preferred, but if you have a source in Russian or Ukranian maybe you can translate the relevant passage?) –jacobolus (t) 06:24, 13 December 2025 (UTC)

I have access to the cited source, but at the moment I cannot quickly locate the exact page. The identity is stated explicitly in the section on standard trigonometric identities. Anyone with access to the book should be able to verify it there.

Russian Wikipedia discussion:

Ukrainian Wikipedia discussion: Геометричне доведення значення Exact trigonometric values#Common angles, ; Обговорення тотожності:

Ambitious miss (talk) 06:51, 13 December 2025 (UTC)

Okay, it's Bronshtein and Semendyayev doi:10.1007/978-3-662-46221-8_2, § 2.7.2.5 "Sum and Difference of Two Trigonometric Functions", Eqs. 2.120–121. I think this is a sufficient reliable source. It would be even better if we had separate articles for the various sections of this article, which could include proofs, diagrams, historical discussion, applications, more obscure variants, etc. –jacobolus (t) 10:30, 13 December 2025 (UTC)

Summarize Timeline Fact Check

Request to Correct Errors in Ukrainian Wikipedia's Exact Value Table

@Jacobolus:, I am writing here because the English Wikipedia article "Exact trigonometric values" is highly accurate and serves as a reliable standard for exact trigonometric values, which is important for cross-language consistency.

I have found a clear error in the Ukrainian Wikipedia article uk:Список тригонометричних тотожностей, specifically in the table section § Таблиці значень тригонометричних функцій (Table of values of trigonometric functions).

The values for ${\displaystyle \sin(\pi /5)}$ (36°) and ${\displaystyle \cos(\pi /10)}$ (18°) are incorrectly assigned in the Ukrainian table.

My specific request is: I propose that a knowledgeable editor or project member discuss this discrepancy on the Ukrainian Wikipedia's talk page, referencing the English Wikipedia's established correct values (e.g., as found in the section 18°, 36° of the English article *Exact trigonometric values*), to ensure the correction is made there.

Thank you for your assistance in maintaining the accuracy of trigonometric content across language projects.

Ambitious miss (talk) 14:01, 13 December 2025 (UTC)

---

Interwiki Consistency Check

I am raising an issue concerning the consistency of trigonometric identities across different language Wikipedias, specifically regarding the conversion of sums/differences involving cotangent and tangent into a product form.

The identities in question are: ${\displaystyle \cot \alpha \pm \tan \beta ={\frac {\cos(\alpha \mp \beta )}{\sin \alpha \cos \beta }}}$

1. Request for Correction on Ukrainian Wikipedia
   The Ukrainian Wikipedia article uk:Список тригонометричних тотожностей in the section § Формули перетворення суми функцій (Formulas for sum conversion), contains an error in the combined notation: $${\displaystyle \tan \alpha \pm \cot \beta ={\frac {\pm \cos(\alpha \mp \beta )}{\cos \alpha \sin \beta }}.}$$ This needs correction on the Ukrainian article.
2. Proposal for Inclusion on Russian Wikipedia
   The Russian Wikipedia article ru:Тригонометрические тождества, in its equivalent section § Формулы преобразования суммы функций, currently omits these useful cot±tan identities entirely. Their inclusion would greatly enhance the completeness of the Russian article.
   Action Requested: I respectfully request that knowledgeable editors consider raising this issue on the respective talk pages of the Ukrainian and Russian articles, referencing the established correct forms to encourage cross-wiki consistency and accuracy.

Thank you for your assistance. Ambitious miss (talk) 14:54, 13 December 2025 (UTC)

These are both topics for discussion at Ukrainian and Russian Wikipedias, not here. Editors at English Wikipedia generally leave other language Wikipedias alone (unless they are speakers of those languages who contribute to both projects). –jacobolus (t) 02:02, 15 December 2025 (UTC)

Under Linear combinations, the addition for the multiple sine functions with the same frequency has a double summation, which seems extremely suspicious. The number of terms is suggested to be O(n^2). Following the link to the linked article Wikipedia "Phasor Addition" shows an more reasonable formula, explaining how to get there using complex exponential notation; and indeed the result is expressed in a linear amount of terms (O(n)), which is what you would expect. The cited article https://mathworld.wolfram.com/HarmonicAdditionTheorem.html seems to agree with the formula on this page, but the formula developed is stated without any proof, while being needlessly verbose. Simply summing the horizontal and vertical components separately for the complex number in A should do the trick, exactly like is done for the phase: arctan(vertical_sum, horizontal_sum). Mcourteaux (talk) 09:58, 14 January 2026 (UTC)

Summarize Timeline Fact Check

Several of the identities without variables are special cases of other identities with variables. Morrie's Law, for instance, is shown as a special case of another identity.

Euler's arctangent identity is presented as a single example ${\displaystyle (\arctan 1/2=\arctan 1/3+\arctan 1/7)}$ when in fact it's a special case of:

${\displaystyle \arctan {\frac {1}{x}}=\arctan {\frac {1}{x+y}}+\arctan {\frac {y}{x^{2}+xy+1}}}$

where ${\displaystyle (x,y)=(2,1).}$ Should the full arctangent identity here be included in the article? It seems to fit with many of the other examples in that section. ES2 (talk) 03:10, 12 March 2026 (UTC)

Euler also wrote it down in this form: https://archive.org/details/commentariiacade09impe/page/231/mode/1up –jacobolus (t) 04:59, 12 March 2026 (UTC)

On the other hand, this is just a straight-forward modification of the arctangent identity mentioned in § Angle sum and difference identities $${\displaystyle \arctan a-\arctan b=\arctan \left({\frac {a-b}{1+ab}}\right),}$$ where we substitute ⁠${\displaystyle a=1/x}$⁠ and ⁠${\displaystyle b=1/(x+y)}$⁠, so that $${\displaystyle {\frac {a-b}{1+ab}}={\frac {{\tfrac {1}{x}}-{\tfrac {1}{x+y}}}{1+{\tfrac {1}{x(x+y)}}}}={\frac {(x+y)-x}{(x^{2}+xy)+1}}.}$$

There are many such modifications, and it may be a bit out of scope to discuss every possibility. –jacobolus (t) 05:14, 12 March 2026 (UTC)

I don't think we would need to discuss every possibility, I'm just thinking about the form Euler wrote it in. ES2 (talk) 18:16, 16 March 2026 (UTC)

Seeing no major objections I've decided to WP:BOLD. ES2 (talk) 18:12, 18 March 2026 (UTC)