Why do negative reciprocals always have a product of - 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.

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.