trigonometry - Proving the angle sum and difference identities for sine and cosine without involving the functions' geometric meanings

For well known identities

$$
\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta
$$
$$
\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta
$$

is it possible to provide a proof which does not involve geometric meaning of sine and cosine functions (that is use of Ptolemy’s theorem)?

Answer

The answer is that it depends how you define the sine and cosine functions; if they have a geometric definition then geometry has to come in somewhere. In fact, they relate closely to the concept of similarity in the Euclidean plane, and are useful in this context because similar triangles have equal angles.

Sine and cosine are also related to the exponential function in the complex plane through the identity $$e^{ia}=\cos a +i\sin a,$$ and we can compute
\begin{align}
e^{i(a+b)}&=e^{ia}e^{ib}\\
\cos (a+b)+i\sin (a+b)&=(\cos a +i\sin a)(\cos b +i\sin b)\\
&=(\cos a\cos b-\sin a \sin b)+i(\sin a \cos b+\cos a\sin b).
\end{align}

Equating real and imaginary parts then gives what we want, and this is applicable generally. Some geometric proofs and constructions apply only to a specific range of values or require considering various cases.