Why is $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$ Why is $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$? How can we find the RHS if we don't know what it is? (instead of proving the identity itself)
I could find a geometric solution in Wikipedia, but is there any solution that doesnt require drawing something?
Edit: I now saw some nice proofs using $e$ and Euler's identity. I would appreciate anything NOT using them too, as a change.
Out of curiosity, Is there a proof that says you actually "can't" prove the identity using only other simple 1-variable identities like $\sin^2\alpha+\cos^2\alpha=1$?

Consider the vectors $$\vec{u}=(\sin \alpha,\cos \alpha)= \left(\cos \left(\frac{\pi}{2}-\alpha\right),\sin\left(\frac{\pi}{2}-\alpha\right)\right) $$
and $$\vec{v}=(\cos \beta,\sin \beta) $$
in $\mathbb R^2$. The dot product $\vec{u} \cdot \vec{v}$ can be calculated in two ways:


*

*$\vec{u} \cdot \vec{v}=\|\vec{u}\| \|\vec{v}\| \cos \theta$
, where $\theta=\frac{\pi}{2}-\alpha-\beta$ is the angle between $\vec{u}$ and $\vec{v}$. Since $\vec{u},\vec{v}$ are unit vectors, this representation gives $\vec{u} \cdot \vec{v}=\cos \left( \frac{\pi}{2}-\alpha-\beta \right)=\sin(\alpha+\beta)$.

*$\vec{u} \cdot \vec{v}=\sin \alpha \cos \beta +\sin \beta \cos \alpha$.
Compare the two.  
