Yes.
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1 is the only positive number that is its own reciprocal. -1 is the negative number that is its own reciprocal.
The reciprocal of -1/10 is -10, and the negative of -10 is 10 .
one slope is the negative reciprocal of the other
Yes.
The cosecant is the reciprocal of the sine function. Now, the reciprocal of a positive number is positive, and the reciprocal of a negative number is negative.