I am quoting from Carl Boyer's _A History of Mathematics_ (1968), p.339:
Vi`ete...in another work a few years later, _Variorum de rebus mathematicis_ (1593), there is a statement equivalent to our law of tangents:... Though Vi`ete may have been the first to use this formula, it was first published by a more obscure figure, Thomas Finck, in 1583 in _Geometria rotundi_. (I've omitted the formula).
I don't know why Boyer thinks that Vi`ete may have been the discoverer of this formula, even though Finck's book is ten years earlier. In the _Norton History of the Mathematical Sciences_, by I.Grattan-Guinness (1997), Finck's book is called _Geometria rotunda_, and it is stated that the names _tangent_ and _secant_ were introduced in that book.