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I don't know if I'm asking for too much, but the proofs I've seen of the statement
$$\sin(x+y) =\sin(x)\cos(y) + \cos(x)\sin(y)$$
consist of drawing a couple of triangles, one on top of each other and then figuring out some angles and lengths until they arrive at the identity.
And I agree with the proof, is just that, even by flipping the triangle around, it only proves the identity for the case $x+y<\pi/2$, or if it does prove it for all values of $x$ and $y$, I wouldn't understand why.
As to construing a proof by using Euler's identity or the derivatives of sin and cos, I would ask the writer to first prove his/her already accepted formulas without using the addition identity.
So that is my humble question. How could one prove that for all the values of $x$ and $y$, the identity $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$ holds.
Any thoughts/ideas would be really appreciated.