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  1. The real secret to ‘improve’ your fluency is to enjoy the learning process.

  2. The real secret to ‘improving’ your fluency is to enjoy the learning process.

Which one is correct here?

And if the second one or both of them are correct, why is that? As far as I know, we use infinitive verbs(like: do, play, improve etc.) after ‘to’.

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Improving. You are not using an infinitive construction. 'to' is a preposition taking you to a noun. In this case improving is a gerund which can be thought of as a noun in this context.

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You have asked a question about licensing, i.e., which words (in this case secret) permit which grammatical constructs as complements (those structures that complete the meaning of the words). In this case, secret licenses the following

[1a] secret to-infinitive
[1b] secret prepositional phrase

Thus you can say

[2a] the secret to improve your fluency

or indeed use any apt infinitive:

[3a] the secret to sharpen your fluency

But notice that secret does not license a bare infinitive:

[4a]* the secret sharpen your fluency

In the case of the prepositional phrase, consider

[2b] the secret to my heart
[3b] the secret about their marriage
[4b] the secret in the hieroglyphics

In the three examples above, the objects of the prepositions are are the simple noun phrases my heart, their marriage, and the hieroglyphics, but nothing prevents the appearance of a gerund, i.e., a present participle that functions as an object:

[5b] the secret to improving fluency

In the [a] cases, the word to is part of an infinitive; in the [b] cases, it's a preposition, and any apt preposition will do:

[6b] the secret for improving fluency

Alas, licensing is part of the idiom of the language, and as such it's almost impossible to say why certain words license certain constructs and not others.

  • So, in the [b] cases they are plural complements/compound nouns. Right? – Strider Jan 7 at 10:22
  • [2b] and [3b] are singular. [4b] is plural. A compound noun is made up of two words like heartstrings. Do you mean a compound object? That's two objects joined by a conjunction like my heart and soul. The number (singular or plural) of the complements here doesn't affect the grammar. I'm afraid I'm not answering your question. – user105719 Jan 7 at 10:37

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