Are the two carbon atoms of C2H6 equivalent? Briefly explain the evidence for your answer.?

The structure of C₂H₆ is shown in the figure below.

The two carbon atoms of C₂H₆ are equivalent.

This can be explained as the bond lengths of the six C-H bonds are all equal to 109.5 pm. If they are not equivalent, the C-H bond lengths on the two sides will not be equal. Also, this can be explained by other symmetric properties such as equal bond angles.

if you were given two images with one the mirror image of the other, could you say which one was the original (could you even tell if they were a mirror image or direct copy?).  If we were to randomly spin the molecule, could you tell which carbon was on which side before spinning?  No.  Therefore, they are equivalent, identical in all ways that we can use to identify them as unique (unless there are isotopic variations).

Yes, because the molecule is symmetrical :)