1. Conversión a senos y cosenos:
Sea $x = \alpha + \beta - \gamma$ y $y = \alpha - \beta + \gamma$.
$$ \frac{\sin x \cos y}{\cos x \sin y} = \frac{\sin \gamma \cos \beta}{\cos \gamma \sin \beta} $$

2. Aplicación de componendo y dividendo:
$\frac{a}{b} = \frac{c}{d} \implies \frac{a+b}{a-b} = \frac{c+d}{c-d}$
$$ \frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} = \frac{\sin \gamma \cos \beta + \cos \gamma \sin \beta}{\sin \gamma \cos \beta - \cos \gamma \sin \beta} $$
$$ \frac{\sin(x+y)}{\sin(x-y)} = \frac{\sin(\gamma+\beta)}{\sin(\gamma-\beta)} $$

3. Sustitución de variables:
$x+y = 2\alpha$, $x-y = 2\beta - 2\gamma$.
$$ \frac{\sin 2\alpha}{\sin(2\beta - 2\gamma)} = \frac{\sin(\beta + \gamma)}{\sin(\gamma - \beta)} $$
Como $\sin(\gamma - \beta) = -\sin(\beta - \gamma)$ y $\sin(2\beta - 2\gamma) = 2\sin(\beta - \gamma)\cos(\beta - \gamma)$:
$$ \frac{\sin 2\alpha}{2\sin(\beta - \gamma)\cos(\beta - \gamma)} = \frac{\sin(\beta + \gamma)}{-\sin(\beta - \gamma)} $$

4. Análisis de casos:
Si $\sin(\beta - \gamma) = 0$, se cumple la igualdad.
Si no, simplificamos y multiplicamos:
$$ -\sin 2\alpha = 2\cos(\beta - \gamma)\sin(\beta + \gamma) $$
Aplicando $2\sin A \cos B = \sin(A+B) + \sin(A-B)$:
$$ -\sin 2\alpha = \sin(2\beta) + \sin(2\gamma) \implies \sin 2\alpha + \sin 2\beta + \sin 2\gamma = 0 $$

$$ \boxed{\sin(\beta - \gamma) = 0 \lor \sin 2\alpha + \sin 2\beta + \sin 2\gamma = 0} $$