Is c = d ? (1) cd = 1 (2) |c| = d

Statement 1 is not sufficient, because maybe c = d = 1, or maybe c = 1/2 and d = 2. Statement 2 is not sufficient because maybe c = d = 1, or maybe c = -1 and d = 1, among other possibilities.

Using both Statements, if cd = 1, then cd is positive, and c and d have the same sign. But we also know d = |c|, so d equals the absolute value of something nonzero, and d must be positive. Since c and d have the same sign, c is also positive, and if c is positive, |c| = c. So Statement 2 tells us d = c, and the answer is C (or you could notice Statement 2 tells you that either d = c or d = -c, and notice only the "c = d" case works with Statement 1, since in the c = -d case, we don't get any real number solutions).




Here, I think you may have done something many people do, especially in more abstract DS questions -- I think you may have assumed, when you used both Statements, that the answer to the question "Is c = d?" is "yes", and then tried to prove whether Statement 2 was true or false. But any time you do that, you're doing DS in the wrong direction -- in DS, the Statements are facts, so they can never be false. If you invent a numerical example that makes a Statement false, then it's an example that is simply not relevant to the problem, and you'll need to look elsewhere for valid examples (or you might try to solve the problem in a different way).

What we don't know here is the answer to the original question "is c = d?", and you don't want to assume the answer to that question in advance (that's technically what is known as "begging the question" in logic, though that phrase often has the unrelated meaning "raising the question" in everyday speech). If you were hoping to prove the answer is E here by inventing numerical examples, your goal is to try to come up with one numerical example that works with both Statements, and that makes the answer to the original question "yes" (the only such example is c = d = 1 in this question), and one numerical example that works with both Statements, and that makes the answer to the original question "no" (so here you'd be looking for numbers where c is not equal to d, but which work with both Statements -- if you could do that here the answer would definitely be E, but in this particular problem, that turns out to be impossible, so the answer is C; the answer to the original question must be "yes" when we use both Statements).