(sin(B − C)) / (sin(B + C)) = (b² − c²) / a²

sin(A)/a = sin(B)/b = sin(C)/c = k

⇒ sinA = ka, sinB = kb, sinC = kc

L.H.S. = sin(B − C) / sin(B + C)

= (sin(B) × cos(C) − cos(B) × sin(C)) / (sin(B) × cos(C) + cos(B) × sin(C))

= [kb·((a² + b² − c²)/(2ab)) − kc((a² + c² − b²)/(2ac))] / [kb((a² + b² − c²)/(2ab)) + kc((a² + c² − b²)/(2ac))]

= [k((a² + b² − c²)/(2a²)) − k((a² + c² − b²)/(2a²))] / [k((a² + b² − c²)/(2a²)) + k((a² + c² − b²)/(2a²))]

= [(a² + b² − c²) − (a² + c² − b²)] / (2a²)

= [2(b² − c²)] / (2a²)

= (b² − c²) / a²