The correct option is "in public" (A) because it is the most appropriate phrase to complete the sentence. "In public" is the correct way to express the idea that Inyang has always been shy to speak in front of other people or in a public setting.

The other options are not correct:

"Publicly" (B) is an adverb, but it does not fit the sentence structure as well as "in public" does.
"In the public" (C) is grammatically incorrect. It should be "in public" without the article "the."
"With the public" (D) does not convey the intended meaning of being shy to speak in front of others.