Verified answer

I will add to the above answer.

The equation is only true when e^(-x) sin(2x) is 0. Since e^(-x) cannot be 0, sin (2x) must be 0.

Sin(2x) is 0 when x is ...-π,-0.5π,0,0.5π,π...

So it is true for all x=0.5π*n where n is an integer.

No, the two sides are not equal. The right side is the first term of the expansion of the left side. The right side also needs the term

+ e^(-x) sin(2x)

to have equality.