There really isn't any trigonometry here apart from sine rule, and because sine rule is derived from similar triangles, anything you can say using sine rule, you can also say using similar triangles.

The trigonometry thing is simply a marketing gimmick for this proof. There is no more or less trigonometry in this proof than there is in Einstein's proof. In fact, you can just taken Einstein's construction and reformulated that proof in their language by using sine rule instead of similar triangles. But then the gimmick would be too obvious.