Let's look at the structure of these two sentences and how they work:

• "I believe him to be a liar."
• "I believe him a liar."

Both sentences are correct, but they follow slightly different grammar patterns.

The first sentence, "I believe him to be a liar," is commonly used in modern English. It follows the pattern "believe + object + to be + complement." This makes the meaning very clear: you are convinced that he is a liar.

The second sentence, "I believe him a liar," is a shorter and older way of saying the same thing. Instead of using "to be," the noun "liar" acts directly as the complement of "him." While this structure is grammatically correct, it is less common in everyday speech today. You might see it in literature or formal writing.

• In modern English, people usually say "I believe him to be a liar." It sounds more natural.
• "I believe him a liar" is correct but less common. It may sound old-fashioned or poetic.

Many traditional grammar books, especially older ones, explain these structures in detail and separate noun complements from infinitive complements. However, modern English books focus more on how people actually speak and write today.

The key takeaway is: both sentences are correct, but modern English is flexible. The most important thing is to speak and write in a way that feels natural and clear.