The correct answer is: E. No error
The sentence is grammatically correct. The word “unintentionally” is an adverb that modifies the verb “let.” The word “out” is a preposition that shows the direction of the action. The word “secret” is the direct object of the verb “let.” The word “by” is a preposition that introduces the phrase “talking about it loudly in the library.” The word “it” is a pronoun that refers to the secret. The word “loudly” is an adverb that modifies the verb “talking.” The word “in the library” is a prepositional phrase that modifies the verb “talking.”
The sentence is clear and concise. It conveys the intended meaning without any ambiguity. The sentence is also consistent with the rules of grammar.