"The man is cooking."
- La is the feminine definite article
- homme is a masculine noun
- The h at the beginning of homme is silent so the word is pronounced as if starting with a vowel → o, therefore
- L'homme = "The man"
Btw, there is no need to capitalise cuisine. ☺
Question: Did you select la because the noun cuisine is feminine? In the given sentence, cuisine is a verb - the 3rd person singular present indicative conjugation of the infinitive cuisiner → "to cook".
faire la cuisine means "to prepare a meal" which one could do without actually "cooking" (make a salad, a sandwich etc).
"Faire cuire" is basically to cook (meat etc...), and
"cuisiner" alone also means to cook (make lunch or dinner).
"Faire cuisiner" does not make any sense, except if you have someone cook in your place, but you would probably say it another way.
to continue that thought...
l'homme cuit = the man is cooking, as in his skin is cooking
l'homme cuit le poulet = the man is cooking the chicken, as in he himself is the oven that heats up the chicken (maybe he is Superman)
l'homme cuit le poulet dans une casserole = the man is cooking the chicken in a pan... (this works, but still could insinuate that he is heating up the pan magically)
*faire + infinitive" = to have something done
l'homme fait cuire le poulet dans une casserole = the man has the chicken cooked in a pan, the man cooks the chicken in a pan
l'homme fait cuire le poulet = the man has the chicken cooked (in a pan or oven etc), same thing for faire bouillir, faire griller, faire frire, faire cuire à la vapeur, etc
l'homme fait la cuisine = the man makes / prepares the meal
l'homme cuisine = the man cooks (prepares the meal, heats it up, etc)
l'homme fait cuisiner sa femme = the man has his wife cook (prepare the meal, heat it up, etc)