Use the complex exponential to derive trigonometric identities such as cos(0 + v) = cos cos - sin sin, and then show that 2 sin sin 2 sin cos 4 = = cos(0) - cos(0 + y), sin(0 + y) + sin(0 - y). This calculation connects the solution given by d'Alembert in terms of traveling waves and the solution in terms of superposition of standing waves.