I've been running my fingers through the internet and books probing why does the triangle law of vector addition works, but I've failed to find a satisfactory answer, rather, any answer at all, for that matter.
Where ever the law is mentioned, they consider finding $R = √(A² + B² + 2AB \cos\theta)$ and $\tan\alpha$ to be the proofs the law, however I am unable to understand why does the law work? Is there a mathematical proof that the third side of the triangle will always actually be the the sum of other two vectors?
Why does the triangle law of vector addition work, at all?
The triangle law follows directly from the defining axioms of vectors*. Suppose you have three vectors such that $\vec a + \vec b = \vec c$. Then by the axioms: $$\vec a + \vec b + (-\vec c) = \vec c + (-\vec c)$$ $$\vec a + \vec b + (-\vec c) =0$$
This means that the three vectors form a closed figure. If a series of line segments has a net displacement of zero that means that the path returns back where it started. Any series of line segments that ends up back where it started is a closed figure.
A closed figure with three sides is a triangle. Hence the resultant of adding two vectors is the third side of the triangle formed by the endpoints of the summed vectors.
*A good review of the axioms of vectors is here: https://www.math.ucla.edu/~tao/resource/general/121.1.00s/vector_axioms.html Note that this proof uses only the definition of addition, the definition of the inverse, and the axioms of addition.