I'm new to this stack community, please bear with me as I try to explain my question properly. Recently I came across with these trigonometric identities (where $ \omega + \phi + \psi = 180^\circ $):
$$\tan \omega + \tan \phi + \tan \psi = \tan \omega \tan \phi \tan \psi$$
$$\sin (2\psi ) + \sin (2 \phi) + \sin (2 \omega) = 4 \sin \omega \sin \phi \sin \psi$$
and a user who answered the first one mentioned that such identity belonged to the 18th century scientist Antonio Cagnoli. I tried to look for sources regarding this but I only could find the wiki entry and this other entry in World cat. The OP of the answer had referenced his proof by this book and it is consistent with what it appears on World cat under:
Trigonométrie rectiligne et sphérique by Antonio Cagnoli( Book )
18 editions published between 1808 and 2017 in French
and Undetermined and held by 117 WorldCat member libraries worldwide
Although it does mention in the book about the identity with the tangent function it doesn't specifically say about the one with the double angle sine function, which is the one that has triggered my attention the most.
Therefore my question arises that other than this source (the book mentioned) are any others out there which can reassure or tell if these two belong to Cagnoli?.
Initially I wanted to post this question in the Mathstack site but I felt this place would be more proper. I'm looking forward for your answers.