Perceba que $P = (sin \ ax + sin \ bx)(sin \ ax - sin \ bx)$. Utilizando as fórmulas de Werner, obtemos: $$P = \left[2 sin\left(\dfrac{ax + bx}{2}\right) cos\left(\dfrac{ax-bx}{2}\right)\right]\left[2 sin\left(\dfrac{ax - bx}{2}\right) cos\left(\dfrac{ax + bx}{2}\right)\right]$$ $$P = \left[2 sin\left(\dfrac{ax + bx}{2}\right) cos\left(\dfrac{ax + bx}{2}\right)\right]\left[2 sin\left(\dfrac{ax - bx}{2}\right) cos\left(\dfrac{ax - bx}{2}\right)\right]$$ $$P = [sin \ (a + b)x] \cdot [sin \ (a - b)x]$$