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,

1. How can I prove that these formulas are correct?
2. 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.

Here are my favorite diagrams:

As given, the diagrams put certain restrictions on the angles involved: neither angle, nor their sum, can be larger than 90 degrees; and neither angle, nor their difference, can be negative. The diagrams can be adjusted, however, to push beyond these limits. (See, for instance, this answer.)

Here’s a bonus mnemonic cheer (which probably isn’t as exciting to read as to hear):

Sine, Cosine, Sign, Cosine, Sine!
Cosine, Cosine, Co-Sign, Sine, Sine!

The first line encapsulates the sine formulas; the second, cosine. Just drop the angles in (in order $$\alpha$$, $$\beta$$, $$\alpha$$, $$\beta$$ in each line), and know that “Sign” means to use the same sign as in the compound argument (“+” for angle sum, “-” for angle difference), while “Co-Sign” means to use the opposite sign.