Looking for an alternative proof of the angle difference expansion I have thought about this for a while and have no progress.
Does there exist a purely Euclidean Geometric proof of the Angle Difference expansion for Sine and Cosine, for Obtuse angles?
 A: Here's a hint, in the form of adapting my Angle-Sum diagram to a couple of obtuse cases. Perhaps they'll guide you to adapting my Angle-Difference diagram appropriately.

Non-Obtuse $\alpha$ and $\beta$, with Obtuse $\alpha+\beta$:
 
$$\begin{align}
\phantom{|}\sin(\alpha+\beta)\phantom{|} &= \sin\alpha \cos \beta + \cos\alpha \sin \beta \\[6pt]
|\cos(\alpha+\beta)| &= \sin\alpha \sin\beta - \cos\alpha \cos\beta \\
\to\qquad \phantom{|}\cos(\alpha+\beta)\phantom{|} &= \cos\alpha \cos\beta - \sin\alpha \sin\beta 
\end{align}$$

Non-Obtuse $\alpha$, with Obtuse $\beta$ and $\alpha+\beta \leq 180^\circ$:

$$\begin{align}
\phantom{|}\sin(\alpha+\beta)\phantom{|} &= \cos\alpha\sin\beta - \sin\alpha\,|\cos\beta| \\
\to\quad \phantom{|}\sin(\alpha+\beta)\phantom{|} &= \sin\alpha \cos\beta + \cos\alpha \sin\beta \\[6pt]
|\cos(\alpha+\beta)| &= \cos\alpha\,|\cos\beta| + \sin\alpha \sin\beta \\
\to\quad \phantom{|}\cos(\alpha+\beta)\phantom{|} &= \cos\alpha\cos\beta - \sin\alpha\sin\beta
\end{align}$$

The cases for $\alpha+\beta > 180^\circ$ are left as exercises to the reader.
