Do the angle addition identities only work for positive angles? The derivations of the angle addition identities that I have seen depend on right triangles so do these identities still work if the angles are negative, as allowed for angles on the unit circle? The angle addition identities that I am talking about are here: https://derive-it.com/2020/12/12/derivation-10-angle-addition-identities/
For example, what if one of the angles is negative?
 A: The Angle Addition Identities were derived using $\sin\theta \equiv \frac{opp}{hyp}$ and $\cos\theta \equiv \frac{adj}{hyp}$. In addition, $\theta$ was constrained to the interval of $[0, \frac{\pi}{2})$.
The question is, do the Angle Addition Identities work if the arguments of the sine function and cosine functions are generalized to include negative numbers? In the following, $\theta$ will still be constrained to the interval of $[0, \frac{\pi}{2})$. The generalization of choice is $ \displaystyle \sin(-\theta) = -\sin\theta$ and $ \displaystyle \cos(-\theta) = \cos\theta$ This generalization is applicable to the unit circle, specifically in the 1st and 4th quadrants.
The Angle Addition Identities are

*

*$\displaystyle \sin(x+y) = \sin x \cos y + \sin y \cos x$


*$\displaystyle \cos(x+y) = \cos x \cos y - \sin x \sin y$.
Next use the above generalizations of the sine and cosine functions. There are three possible cases in which at least one of the input angles is negative.
The first case to investigate is $y \rightarrow -y$ while maintaining $y \in [0,\frac{\pi}{2})$:

*

*$\displaystyle \sin(x+(-y)) \stackrel{?}{=} \sin x \cos (-y) + \sin (-y) \cos x$


*$\displaystyle \cos(x+(-y)) \stackrel{?}{=} \cos x \cos (-y) - \sin x \sin (-y)$.
The second case to investigate is $x \rightarrow -x$ while maintaining $x \in [0,\frac{\pi}{2})$:

*

*$\displaystyle \sin((-x)+y) \stackrel{?}{=} \sin (-x) \cos y + \sin y \cos (-x) $


*$\displaystyle \cos((-x)+y) \stackrel{?}{=} \cos (-x) \cos y - \sin (-x) \sin y $.
The third case and final case to investigate is $x \rightarrow -x$ while maintaining $x \in [0,\frac{\pi}{2})$ and $y \rightarrow -y$ while maintaining $y \in [0,\frac{\pi}{2})$:

*

*$\displaystyle \sin((-x)+(-y)) \stackrel{?}{=} \sin (-x) \cos (-y) + \sin (-y) \cos (-x) $


*$\displaystyle \cos((-x)+(-y)) \stackrel{?}{=} \cos (-x) \cos (-y) - \sin (-x) \sin (-y) $.
One realization to make is that the form of the previous six "equations" is no different from the original Angle Addition Identities. So the question that remains is whether these identities make geometrical sense. This question can be further investigated with geometry. Three new pictures would be needed along with six separate proofs.
