Final answer:
King Claudius sends Rosencrantz and Guildenstern to spy on Hamlet to discover the reason behind his odd behavior, using their friendship as a way to gain insight and ensure his own safety.
Step-by-step explanation:
King Claudius gives the reason for sending Rosencrantz and Guildenstern to spy on Hamlet under the pretext of their long-standing friendship. He hopes that they will draw out the cause of Hamlet's strange and erratic behavior since the death of his father and the subsequent marriage of his mother, Queen Gertrude, to Claudius. The play suggests that Claudius is concerned about his own safety and the stability of his reign, given Hamlet's potential for vengeance.
In Shakespeare's narrative, the characters of Rosencrantz and Guildenstern are often seen as pawns of the king, used to probe the depths of Hamlet's mind and report back to Claudius. Although the conversation leading to their being sent is not directly present in the example text provided, the action aligns with Claudius's character as a manipulative and self-preserving ruler.