What is the value of $$\frac{{\left( {\sin 4x + \sin 4y} \right)\left[ {\tan \left( {2x - 2y} \right)} \right]}}{{\sin 4x - \sin 4y}}?$$

Solution(By Examveda Team)

$$\eqalign{ & \frac{{\left( {\sin 4x + \sin 4y} \right)\left[ {\tan \left( {2x - 2y} \right)} \right]}}{{\sin 4x - \sin 4y}} \cr & \Rightarrow \frac{{2\sin \left( {\frac{{4x + 4y}}{2}} \right)\cos \left( {\frac{{4x - 4y}}{2}} \right)\left[ {\frac{{\sin \left( {2x - 2y} \right)}}{{\cos \left( {2x - 2y} \right)}}} \right]}}{{2\cos \left( {\frac{{4x + 4y}}{2}} \right)\sin \left( {\frac{{4x - 4y}}{2}} \right)}} \cr & \Rightarrow \frac{{\sin \left( {2x + 2y} \right)\cos \left( {2x - 2y} \right)}}{{\cos \left( {2x + 2y} \right)\sin \left( {2x - 2y} \right)}} \times \left[ {\frac{{\sin \left( {2x - 2y} \right)}}{{\cos \left( {2x - 2y} \right)}}} \right] \cr & \Rightarrow \tan \left( {2x + 2y} \right) \cr} $$