Question

Why do negative reciprocals always have a product of -1?

Answer 1

No matter what 'x' is, the negative reciprocal of 'x' is  -(1/x).

Now let's multiply them:

                               (x) times  -(1/x)  =  -(x/x)  =  -1 .

And there you are.
That's why.

Answer 2

For any number x, when we multiply by -1/x, we get -x/x, or -1.  As we haven't made any assumptions about x (other than x is not equal to zero, in which case x does not have a defined reciprocal), all negative reciprocals have a product of -1.