I am generally dissatisfied with the way trigonometric addition formulas like $$ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) $$ are derived in high school textbooks. There are numerous proofs, some of which are short but unintuitive, some of which introduce unnecessary calculations, many with restrictions on $\alpha$, $\beta$ and $\alpha + \beta$, like having to lie between $0$ and $\frac{\pi}{2}$ radians.
To me, the addition formulas are simply a coordinatization of the observation that composing a rotation by $\alpha$ with a rotation by $\beta$ yields a rotation by $\alpha + \beta$. Hence in my opinion the proper way to prove such a formula would be to observe that $$ \begin{bmatrix} \cos(\alpha + \beta) & -\sin(\alpha + \beta) \\ \sin(\alpha + \beta) & \cos(\alpha + \beta) \end{bmatrix} = \begin{bmatrix} \cos(\beta) & -\sin(\beta) \\ \sin(\beta) & \cos(\beta) \end{bmatrix} \cdot \begin{bmatrix} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \end{bmatrix} $$ then work out the product on the right to obtain the formulas by equating entries. However, such a proof is beyond the scope of a high-school textbook, as high schoolers - if they know about matrices at all - are rarely taught the link between composition of linear transformations and matrix multiplication.
Is there a way to salvage the essence of this proof - that the formulas are merely a way of expressing that a rotation by $\alpha+ \beta$ can be obtained by composing rotations by $\alpha$ and $\beta$ - without using linear algebra? Of course complex numbers are also out of the question.