<< I'm interested in finding a reference in English to the origin and use of the theorem of tangents ..... >>
An elementary book in English that might be of your interest is Eli Maor's "Trigonometric Delights" [Princeton University Press, 1998.] See chapter xii, p. 152. This book is far from being a scholarly work, though Maor takes his job seriously.
"Around 1580 Viete stated a beautiful theorem that, unfortunately, has all but disappeared from today's textbooks: the Law of Tangents. . . . This theorem follows from the Law of Sines . . . but in Viete's time it was regarded as an independent theorem (3).
Note (3) [p. 162] ... As for other discoverers of the Law of Tangents, see David Eugene Smith, __History of Mathematics__ (1925, rpt. New York: Dover, 1958), vol. 2, pp. 611 and 631.
Fran, it goes without saying that _the_ reference (as far as secondary sources are concerned) for this kind of question is Braunmuehl's "Vorlesungen ueber Geschichte der Trigonometrie" (two volumes), published in Leipzig, B.G. Teubner, 1900-03.
Best wishes from excited Montevideo (UY soccer team is trying tonight to get back to its glorious and golden days),