Question

When proving an identity, we can multiply or divide both sides by the same variable expression.

True

False

Answer 1

False.

You may multiply both sides by any expression (it only enlarges the set of true statements), but you may divide both sides only by an expression that is nonzero for the values of the variable under consideration. Dividing by an expression that can be zero can remove valid cases or produce an invalid equivalence.

Example: from x^2 = x dividing by x gives x = 1, which loses the solution x = 0. In identity proofs you must either avoid dividing by quantities that might be zero or explicitly restrict the domain to exclude zeros.