If you cast a card "without paying its mana cost," you can't pay any alternative costs. Notably, you can't cast a card with bestow as an Aura this way. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, you must pay those.

To choose a card in your hand, clearly separate it from the rest of your hand without revealing it.

If the card has {X} in its mana cost, X is 0.

Although you may choose a land card in your hand, you won't be able to play it even if the player guessed wrong.

You cast the card in your hand as part of the resolution of the triggered ability. Timing restrictions based on the card's type (such as creature or sorcery) are ignored. Other restrictions, such as "Cast [this card] only during combat," are not.

If you don't cast the card, either because the player guessed right or because you choose not to cast it, you don't have to reveal it. Your opponent won't know the reason you didn't cast the card.

If the card is a split card, its mana value is determined by the combined mana cost of its two halves. If the player guesses wrong and you cast a split card this way, you may cast either half (or both halves if the card has fuse), no matter what its mana value is.