Actually, this is more of a proof for the trigonometric identity, $$\sin^2\theta + \cos^2\theta = 1.$$
First use right triangle, $ABC$ with angle $C$ being the right angle.
Drop an altitude from point $C$ to line $AB$ making the point $D$.
Let's name the segments: $AB = c$, $AC = b$, and $BC = a$.
Also, $AD = c_1$, $BD = c_2$, so $c_1 + c_2 = c$. Then, $b\cos A = c_1$, $a\cos B = c_2$, so $b\cos A + a\cos B = c$.
Since we know that $\angle A + \angle B = 90^\circ$, we also know that $\cos B = \sin A$.
Thus, $b\cos A + a\sin A = c$, or $$ \frac{b}{c}\,\cos A + \frac{a}{c}\, \sin A = 1. $$
Since $\frac{b}{c} = \cos A$ and $\frac{a}{c} = \sin A$, we have $$ \cos^2 A + \sin^2 A = 1. $$
It is neither right nor wrong , though it is useful to improve basic intuition.
Euclidean Geometry ( with the list of Axioms & Postulates including the Parallel Postulate ) gives the Pythagorean theorem.
Euclidean Geometry [ with Similarity Property & Pythagorean theorem ] gives rise to trigonometry.
[[ Consider $a^2+b^2=c^2$ : Divide by $c^2$ : $(a/c)^2+(b/c)^2=(c/c)^2$ : Write in trigonometric form : $\sin^2 \theta + \cos^2 \theta = 1$ ]]
Hence it is circular logic to say trigonometry can give Pythagorean theorem.
What your thinking has given is the important intuitive concept that trigonometry & Pythagorean theorem are consistent ( or at least not inconsistent ) limiting to this calculation.
Euclidean Geometry ( with Pythagorean theorem & Etc ) & trigonometry ( with the given Identity & other Concepts ) are two sides of the same coin !
ADDENDUM :
Answer above is self-contained & this Section can be skipped in toto. It is mostly going to elaborate what is given above , while responding to certain concerns raised , including those not relevant to OP Post.
Unfortunately , it might turn out very long , which is the unavoidable consequence of the abstract nature of the Discussion.
It has become too long , sounding like some Opinion Piece ! Posting this , I have great hesitation , though I have to keep my promise of responding to the comments.
(1) Elisha Scott Loomis has reviewed centuries of Proof Suggestions & concluded that the trigonometry way is circular , which is the conclusion of the thoughts of hundreds of geometers over the ages , while Jason Zimba is relatively new & not well reviewed. It might be true or there might be loopholes not yet reveled by thorough assessments by hundreds of geometers : Until then , I will withhold judgement on that. When necessary , I will change my stand. I will try to high-light why I am currently not changing my stand.
(2) Let us say we have arbitrary PT Proof like this : "Axioms , Postulates , Consequence1 , Lemma1 , Lemma2 , Consequence2 , therefore $a^2=c^2/2-x$ , $b^2=c^2/2+x$ , adding these we get PyTheo"
Then dividing thought-out by $c^2$ , we get TrigID.
OK , now we want to ignore or deliberate avoid PyTheo , hence we can easily avoid naming it or avoid isolating that fact , to get TrigID like this , using the Exact Same Machinery : "Axioms , Postulates , Consequence1 , Lemma1 , Lemma2 , Consequence2 , therefore $a^2=c^2/2-x$ , $b^2=c^2/2+x$ , now we can divide though-out by $c^2$ to get $\sin^2 \theta = 1/2 - x/c^2$ , $\cos^2 \theta = 1/2 + x/c^2$ , adding these we get TrigID"
Is that really Trig Way or PyTheo in Disguise ( using the term given by user "Intelligenti pauca" ) or Is it PyTheo hiding in plain sight which we want to not see ?
Explaining that Point in other words , say we are given $x+2y=3$ , $4x-5y=6$ & we want to get $x/y$. We can solve the Simultaneous Equations to first get $x$ , then $y$ , then get the ratio. Is it essentially same when we get $y$ first , then $x$ , then the ratio ?
I think it is essentially same. Like-wise here.
(3) Is the given Proof Correct ? Yes , it is Correct. Is it avoiding PT ? No , PT is still there , though somewhat hidden.
The outstanding issues which are not clarified in that Proof :
How can we say all right angle triangles with same angles have same ratios ? ( It is using Similarity of triangles with same angles )
When we say $ \sin A= \cos B$ , we assume $A+B=90^\circ$ : How can we say angles add up to $90^\circ$ ? It is true only in Euclidean geometry that triangles add up to $180^\circ$. We are implicitly using Parallel Postulate ( & more! ) here.
Not many users are aware of Non-Euclidean geometry , who might think it is nonsensical to question whether triangle adds up to $=180^\circ$ which is generally considered obvious. Truth is : there are cases where it is $<180^\circ$ & there are cases where it is $>180^\circ$.
To reduce the size of this Discussion , I will avoid elaborating on the Uniqueness of Point $D$ no matter how or where we have the given triangle [ Euclid does consider such things in various other Cases ]
(4) A comment by user David K says "There are various sequences in which the pieces can be assembled" with which I agree. More-over "Similarity of triangles with equal angles can be proved without having first proved the Pythagorean Theorem and therefore is not necessarily dependent on the Pythagorean Theorem" is also true. But that should be put in the Post , to make it self-contained , showing what was used & what was not used. Only then that we can claim that it is not redundant or not circular or just renaming.
When we assemble a sequence of arguments to generate 4 or 5 "theorems" at once , including PyTheo & TrigID , then we can not say these are derived independently : the line of argument already includes the machinery for PyTheo , we just wrote the outcomes in some order.
(5) A comment by user ultralegend5385 says "no one cares so much about Euclidean geometry or its postulates when doing analysis" , which is true to a certain extent. There are various Geometry Proof Concepts for matching Derivatives with tangent lines & for mean value theorems & for Evaluating Derivative of $\sin(x)$ geometrically/trigonometrically & even for getting the arc length of curves using $(dx)^2+(dy)^2=(ds)^2$ ( I think , there is no escape of this )
The Comment continues with "triangle ratio follows immediately as a theorem, and the other results like the derivative of $\sin(x)$ , or properties like $\sin^2(x)+\cos^2(x)=1$ follow immediately (without using any Euclidean geometry)" , which is not true.
Euclidean geometry has already been used at the lower levels & we ignore it at higher levels.
If not , it will wrongly imply that calculus + analysis is enough to derive Euclidean geometry , analysis supports Euclidean geometry only , thus Parallel Postulate is true , other (Non-Euclidean) geometries are not consistent with calculus + analysis + real numbers !
When we think about analytic ways to Define trig functions , say $\sin_a$ & $\cos_a$ , we have to use geometry as the hidden base. Without using geometric ways like $\sin_g$ & $\cos_g$ , we might think of what Properties we want the analytic functions to have.
Property 1 : $\sin_a^2 x + \cos_a^2 x = 1$
This is satisfied by $\sin_a x = \sqrt x$ , $\cos_a x = \sqrt {1-x}$ , which is not want we want.
Property 2 : Derivatives alternate between $\sin$ & $cos$ , while $y^{[4]}=y$ , that is , 4th Derivative gives back the function.
This is satisfied by $\sin_a x = e^{-x}$ , $\cos_a x = -e^{-x}$ , which is not want we want.
Eventually , we will have to use the original geometry or trigonometry like this :
Property 3 : We want $\sin_a \equiv \sin_g$ & $\cos_a \equiv \cos_g$ , in the Domain $(0,\pi)$
What-ever functions we now consider will have to match. Proof will necessarily involve geometry to show the Equivalence which will be consistent over-all. There is no escaping that. It might be the the Equivalence is hidden in the abstraction at the lower level & not easily visible at the high level.
(6) Comment by user fleablood says "The OPs prove is purely euclidean pre-Pyth. Th. but post similar triangles" , with which I agree.
Comment continues with "It is, in essence, a purely euc geo proof of Pyth. th. ..." , very true , though we have to fill-in the Missing Parts to check whether the Trig Part is just renaming or redundant or circular , to respond to OP Question title.
Comment concludes with "Prefer if such a proof eschewed mentioning trig names as it misleads" , which is what I have excruciatingly typed out here.
I better stop typing now. It is looking like some abstruse Philosophy text , which may well be what this is !