Something I think got overlooked in examining Frodo's whipsawing attitude of trust vs. skepticism toward Strider over the course of their conversation, and how quickly he shifts into trusting Strider as soon as he starts volunteering info about the Black Riders, is this:
Remember that the very first interaction that any of the hobbits has with Strider is when Strider urgently warns Frodo to stop Pippin from making a fool of himself and blowing all of their cover by telling the story of Bilbo's disappearance. That in itself is such a big flag-waving "hey Frodo, I'm totally an ally" moment that I remember when I first read the book as a young teen it took me aback that Strider would start acting like a "rascal" once he has them behind closed doors. Surely he had proved his value and trustworthiness within seconds of Frodo getting within earshot of him!
Remember that the very first interaction that any of the hobbits has with Strider is when Strider urgently warns Frodo to stop Pippin from making a fool of himself and blowing all of their cover by telling the story of Bilbo's disappearance. That in itself is such a big flag-waving "hey Frodo, I'm totally an ally" moment that I remember when I first read the book as a young teen it took me aback that Strider would start acting like a "rascal" once he has them behind closed doors. Surely he had proved his value and trustworthiness within seconds of Frodo getting within earshot of him!