How can I understand and prove the "sum and difference formulas" in trigonometry? The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true.
\begin{align}
\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\
\cos(\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta
\end{align}
So what I want to know is, 


*

*How can I prove that these formulas are correct?

*More importantly, how can I understand these formulas intuitively?


Ideally, I'm looking for answers that make no reference to Calculus, or to Euler's formula, although such answers are still encouraged, for completeness.

Let $\alpha, \beta, \theta \in \mathbb{R}.$ Consider vectors $\vec{P} = (\cos(\alpha), \sin(\alpha)), \vec{Q} = (\cos(\beta), \sin(\beta) ),$ and their rotated versions $\vec{P'} = (\cos(\alpha + \theta), \sin(\alpha + \theta) ), \vec{Q'} = (\cos(\beta + \theta), \sin(\beta + \theta) ).$
Rotations preserve distances, so $PQ = P'Q',$ ie $(\cos(\alpha) - \cos(\beta)) ^2 + (\sin(\alpha) - \sin(\beta) ) ^2$ $= (\cos(\alpha + \theta) - \cos(\beta + \theta) ) ^2 + (\sin(\alpha + \theta) - \sin(\beta + \theta) ) ^2 ,$ ie ${\color{green}{\cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta) = \cos(\alpha + \theta) \cos(\beta + \theta) + \sin(\alpha + \theta) \sin(\beta + \theta)}}.$
This holds for all values of $\alpha, \beta, \theta.$
Setting $\alpha = 0,$ $\cos(\beta) = \cos(\theta) \cos(\beta + \theta) + \sin(\theta) \sin(\theta + \beta).$ Further setting $\lbrace \beta = A+B, \theta = (-B) \rbrace$ gives $\cos(A+B) = \cos(A) \cos(B) - \sin(A) \sin(B).$
Similarly setting $\alpha = \frac{\pi}{2}$ gives $\sin(\beta) = -\sin(\theta) \cos(\beta + \theta) + \cos(\theta) \sin(\beta + \theta),$ and further setting ${ \lbrace \beta = A + B, \theta = (-B) \rbrace }$  gives ${ \sin(A+B) = \sin(A) \cos(B) + \cos(A) \sin(B) }.$
