This, I would argue, is not a trigonometric proof of the PT.
This is a (previously known*) geometric proof of the PT that uses similar triangles where you have used the definitions of of sine and cosine as shorthand for proportions that we could set as equal to each other that we get from similar triangles.
What I mean is you combine x + y = 1, x = u^2 = \cos^2 \alpha, y = v^2 = \sin^2 \alpha to get \cos^2 \alpha + \sin^2 \alpha = 1, but you don’t need the trig functions in there at all. x + y = 1, x = u^2, y = v^2 give us u^2 + v^2 = 1.
The proof does not rely on a result from trigonometry. It’s just a relabeling of the distances in a similar triangles proof with trig functions.
* See proof #8 at Cut The Knot and note the remark about the simpler proof.
In hindsight, there is a simpler proof. Look at the rectangle (1 + 3 + 4). Its long side is, on one hand, plain c, while, on the other hand, it’s a²/c + b²/c and we again have the same identity.
Or, consider this familiar image and the PT proof that goes with it.
Now everywhere replace a with \sin x, b with \cos x, and c with 1. Have you provided a trigonometric proof of the Pythagorean Theorem? I would argue no, for the simple fact that if it were that easy, this thread wouldn’t exist.
I am not about to argue over what is and isn’t a trig proof. All trig functions are ratios of lengths and could be replaced by those ratios.
Historical note. The first trig function was the chord. Put an angle at the center of a circle. The chord of that angle was the ratio of the length of the chord between the two points where the angle intersected the circle to the diameter of the circle. (Or take a circle of radius 1 so the chord is just the lenght of the chord.) The chord was twice the sine of half the angle. Some genius figured out that the sine function was more useful. Probably because of the \sin(\alpha + \beta) formula.
Probably not; the development of the sine function seems to have happened sometime around the start of the Common Era in India, based on modification of the Hellenistic chord concept. The probable motivation was the superior convenience of tabulating lengths of the side opposite a given angle in a right triangle whose hypotenuse is scaled to the standard-circle radius rather than its diameter.
Namely, a chord table relates a given angle at the center of a circle, as you note, to the length of the side opposite that angle in an isosceles triangle whose equal sides are radii of the circle. If you want to solve an arbitrary right triangle, which is what ancient applied trigonometry was most concerned with, you have to imagine that right triangle as inscribed in a semicircle (hypotenuse equals diameter) with the known angle at the circumference. Then the desired opposite side is the chord of the angle at the center that is double the known angle.
If instead you tabulate half-chords (sines) rather than chords, then you can imagine your right triangle as inscribed in a quadrant (hypotenuse equals radius), with the known angle at the center. And you can read the value of the desired opposite side directly from the table; no doubling of the angle required before lookup.
Trig identities for sines of sums and differences of angles, etc., weren’t developed till centuries later.
