expectorate vs cough

expectorate is not a noun while cough is a noun.

Word	Noun	Adjective	Verb	Adverb
expectorate	No	No	Yes	No
cough	Yes	No	Yes	No

As verbs, cough is a hyponym of expectorate; that is, cough is a word with a more specific, narrower meaning than expectorate:

expectorate: discharge (phlegm or sputum) from the lungs and out of the mouth
cough: exhale abruptly, as when one has a chest cold or congestion

Other hyponyms of expectorate include ptyalise, ptyalize, spew, spit, spue, spit out, splutter, sputter.

expectorate (noun)	cough (noun)
	a sudden noisy expulsion of air from the lungs that clears the air passages; a common symptom of upper respiratory infection or bronchitis or pneumonia or tuberculosis

expectorate (verb)	cough (verb)
discharge (phlegm or sputum) from the lungs and out of the mouth	exhale abruptly, as when one has a chest cold or congestion
clear out the chest and lungs