shorten vs cut down

shorten and cut down both are verbs.

Word	Noun	Adjective	Verb	Adverb
shorten	No	No	Yes	No
cut down	No	No	Yes	No

As verbs, cut down is a hypernym of shorten; that is, cut down is a word with a broader meaning than shorten:

shorten: make shorter than originally intended; reduce or retrench in length or duration
cut down: cut down on; make a reduction in

Other hypernyms of shorten include bring down, cut, cut back, reduce, trim, trim back, trim down.

shorten (verb)	cut down (verb)
become short or shorter	cut down on; make a reduction in
make shorter than originally intended; reduce or retrench in length or duration	intercept (a player)
reduce in scope while retaining essential elements	cause to come or go down
edit by omitting or modifying parts considered indelicate	cut with a blade or mower
make short or shorter	cause to fall by or as if by delivering a blow
	cut with sweeping strokes; as with an ax or machete