What is the value of ( 4x + 4y ) [ ( 2x - 2y ) ] 4x - 4y?

Answer: Option D

$$\begin{array}{rl}& \frac{\left(\sin 4x+\sin 4y\right)\left[\tan \left(2x-2y\right)\right]}{\sin 4x-\sin 4y}\\ & \Rightarrow \frac{2\sin \left(\frac{4x+4y}{2}\right)\cos \left(\frac{4x-4y}{2}\right)\left[\frac{\sin \left(2x-2y\right)}{\cos \left(2x-2y\right)}\right]}{2\cos \left(\frac{4x+4y}{2}\right)\sin \left(\frac{4x-4y}{2}\right)}\\ & \Rightarrow \frac{\sin \left(2x+2y\right)\cos \left(2x-2y\right)}{\cos \left(2x+2y\right)\sin \left(2x-2y\right)}\times \left[\frac{\sin \left(2x-2y\right)}{\cos \left(2x-2y\right)}\right]\\ & \Rightarrow \tan \left(2x+2y\right)\end{array}$$