# Original source of these 2 trigonometric identities

I am interested in knowing what is the original source/author of the following identities:

$$\tan{\frac{\alpha}{2}}\tan{\frac{\beta}{2}}+\tan{\frac{\alpha}{2}}\tan{\frac{\gamma}{2}}+\tan{\frac{\beta}{2}}\tan{\frac{\gamma}{2}}=1\tag{1}.$$

$$\cot{\frac{\alpha}{2}}\cot{\frac{\beta}{2}}\cot{\frac{\gamma}{2}}=\cot{\frac{\alpha}{2}}+\cot{\frac{\beta}{2}}+\cot{\frac{\gamma}{2}}\tag{2}.$$

Where $$\alpha$$, $$\beta$$ and $$\gamma$$ are the angles of a triangle.

I ask this question because Wikipedia does not cite sources or mention the original author.

• @njuffa: As you can see here, I am already aware of that proof. My question is who is the author. Dec 9, 2022 at 11:42
• I think this question is somehow related to my question. Dec 9, 2022 at 15:46
• @njuffa: Thank you very much! You have definitely answered my question. At least now I have the sources. Dec 10, 2022 at 2:02

After hours of searching through scans of books from the 18th and 19th centuries in English, German, French, Italian, Spanish, and Latin I managed to find a source for each of the two identities.

The first one appeared in a list of trigonometric identities for the oblique triangle in a book by the Belgian mathematician Émile Gelin, Éléments de Trigonométrie Plane et Sphérique. Namur: Wesmael-Charlier 1888, p. 106:

[151] [...] $$\mathrm{tg}\frac{1}{2}\mathrm{A} \ \mathrm{tg}\frac{1}{2}\mathrm{B} + \mathrm{tg}\frac{1}{2}\mathrm{B} \ \mathrm{tg}\frac{1}{2}\mathrm{C} + \mathrm{tg}\frac{1}{2}\mathrm{C} \ \mathrm{tg}\frac{1}{2}\mathrm{A} = 1;$$

The second one appeared as an exercise in Joseph Alfred Serre, Traité de Trigonometrié. Paris: Bachelier 1850, p. 67:

2. Démontrer que si $$a,b,c$$ sont trois arcs dont la somme est égale à $$\pi$$, on a:
1°. $$\mathrm{cot}\frac{a}{2} + \mathrm{cot}\frac{b}{2} + \mathrm{cot}\frac{c}{2} = \mathrm{cot}\frac{a}{2} \ \mathrm{cot}\frac{b}{2} \ \mathrm{cot}\frac{c}{2};$$

I think it is unlikely that these are the first (original) instances of these two identities in the literature. The inscribed circle of a triangle is already found in Euclid IV. 4. In the first century C.E., Heron of Alexandria already showed that the area of the triangle is $$\sqrt{s(s – a)(s – b)(s – c)}$$, where the semi-perimeter $$s = \frac{a+b+c}{2}$$. As far a I know, the ancient Greeks already had the notion of a (co-)tangent, as the length of a shadow cast by a gnomon hit by the sun at a particular angle.

I do not know who first worked out that the radius of the inscribed circle is $$r = \sqrt{\frac{(s – a)(s – b)(s – c)}{s}}$$ and that $$\tan\frac{A}{2}=\frac{r}{s-a}, \tan\frac{B}{2}=\frac{r}{s-b}, \tan\frac{C}{2}=\frac{r}{s-c}$$. With modest effort this might lead one to the first formula, for example. I would be very surprised if this did not happen long before the 19th century.

Thinking up trigonometric identities for the oblique triangle for use in exercises and exams or as mathematical puzzles for certain journals appears to have been a favorite pastime of 19th century mathematicians (or at least this is the impression I got from my lengthy perusal of the literature) and this might be the reason such identities are most readily found in books from that time period.

• If you're up for many more hours of searching (probably it'll be months $\ldots),$ there's the listing of papers in the Formulae of Plane Trigonometry sub-section (pp. 463-466) of the Trigonometry, plane and spherical section of Royal Society of London Catalogue of Scientific Papers 1800-1900. Volume I, Pure Mathematics. Also, I recall once coming across a HUGE list of trig identities for triangles published (maybe in 2 or 3 parts) somewhere (early issues of Annals of Math.?) around 1880s to 1900 or so. Dec 10, 2022 at 15:40
• I've got 1850-1921, but apparently I didn't know this in April 2011, according to my reference to his book here. It's possible the birth year I later came across -- not sure when, but I was definitely using 1850 by May 2018 -- is not correct, and 1851 is correct. However, this reference gives 1850. What are your references for 1851? Dec 10, 2022 at 19:54
• (1) Municipality in his place of birth named a street after him and says born 1851. (2) Revue générale, Issues 1-5, 1992 Google snippet says 1851-1921. (3) This one says 1850-1921 in Appendix 1: Robin Onno Buning, "Henricus Reneri (1593-1639) Descartes’ Quartermaster in Aristotelian Territory." PhD dissertation, University of Utrecht 2013. Dec 10, 2022 at 20:02
• One interesting tidbit about Gelin that I found in a 1916 report on the libraries of German-occupied Belgium is that Abbe Emile Gelin in Huy had a private library of some 5000 volumes, making it one of the largest private libraries in the country. Fritz Milkau, "Das Kriegsschicksal der belgischen Bibliotheken", Zentralblatt für Bibliothekswesen, Vol. 33, Nos. 1&2, Jan./Feb. 1916, pp. 1-27 (scan) Dec 10, 2022 at 20:16