Original source of these 2 trigonometric identities

Question

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.

Answer 1

After hours of searching through scans of books from the 18^th and 19^th 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 19^th 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 19^th 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.