Rotate a straight line by angle \(\theta\) from the origin with a starting coordinate:

1. \((1,0) \rightarrow (\cos \theta, \sin \theta)\)
2. \((0,1) \rightarrow (-\sin \theta, \cos \theta)\)
3. \((x,0) \rightarrow (x\cos \theta, x\sin \theta)\)
4. \((0,y) \rightarrow (-y\sin \theta, y\cos \theta)\)
5. \((x,y) \rightarrow (x\cos \theta - y\sin \theta, x\sin \theta + y\cos \theta)\)

Therefore, if we rotate \((1,0)\) first by \(\alpha\) and then by \(\beta\), we would end up first at \((\cos \alpha, \sin \alpha)\), and then at

\( (\cos \alpha \cos \beta - \sin \alpha \sin \beta,   \cos \alpha \sin \beta + \sin \alpha \cos \beta) \)

On the other hand, if we rotate \((1,0)\) by \(\alpha + \beta\), we would end up at

\( (\cos(\alpha + \beta),   \sin(\alpha + \beta)) \)

This means:

\[\begin{aligned} \cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \sin(\alpha + \beta) &= \cos \alpha \sin \beta + \sin \alpha \cos \beta \end{aligned} \]