Zoran Skoda Ptolemy from addition

Want to prove

$$sin(\alpha+\beta) sin(\beta+\gamma) = sin\alpha sin\gamma+sin\beta sin\delta.$$

from the addition formulas. We know $\delta = \pi-(\alpha+\beta+\gamma)$, so with $sin\alpha = a$, $sin\beta = b$ and $sin\gamma = c$ and $cos\alpha = a'$ etc.

$$sin\delta = a b' c' - a b c + a' b'c + a' b c'$$

$$sin(\alpha+\beta) = a b' + a' b$$

$$sin(\beta+\gamma) = b c' + c' b$$

Thus, LHS

$$(a b' + a' b)(b c' + b' c) = a b b' c' + a b' b' c + a'b b c' + a' b b' c$$

and RHS

$$a c + b (a b' c' - a b c + a' b'c + a' b c') = a (1- b b) c + a b b' c' + a' b b' c + a' b b c'$$

The two are the same thanks to the basic trigonometric identity $1-b b = b' b'$.