[He should kill him now and take his Note. He could hardly believe in time travel because this man claimed it was true. No, he wouldn't have even been the first person to claim it was true, as it was a popular enough conspiracy theory. The man was right enough that expression of sentiment wasn't a good argument. With his inability to properly reason and assess reality, he was a danger to Light and himself. No, he was a danger to the world, as any danger to Kira would be. So, he should die.

He wouldn't. Not yet. First, there were a few tests. There were a few other likelihoods Light wanted to explore before making that decision. The first of which was to confirm he was on the right side. He had a hard time believing any enemy of his would come up with such a story - or that they could have gotten his information - but he could at least rule out his relationship with the law.]

Make use of it. [Even being undercover wasn't a good enough reason for a policeman to kill someone.] If you're really who you say you are and can do as you'd like me to believe, you'll make use of your tools.