If Two Numbers Are Opposite Of Each Other, Are Their Reciprocals Also Opposite Of Each Other?

No. Opposites are just the positive and negative of the same number. For example, -4 is the opposite of 4 ( also think of it as in the same distance to 0 on a number line)  Reciprocals on the other hand refer to the switching of the numerator and the denominator in a given fraction. For example 5/3 is the reciprocal of 3/5 ( reciprocals usually used in dividing fractions)
 
So take 3/5    its opposite is -3/5     its reciprocal is 5/3

Let n be the number we're talking about. Then -n is opposite n.

The reciprocal of n is 1/n, and the opposite of that is -1/n, which is the same as 1/(-n).

So, two (different) opposite numbers have opposite reciprocals.

(The number and its opposite must be different for your statement to be true, because if they're not, -n = n = 0, and the reciprocal 1/0 is not defined.)