early 14c., "margin of the land;" earlier "rib as a part of the body" (early 12c.), from Old French coste "rib, side, flank; slope, incline;" later "coast, shore" (12c., Modern French côte), from Latin costa "a rib," perhaps related to a root word for "bone" (compare Old Church Slavonic kosti "bone," and PIE root *ost-), but de Vaan dismisses this and calls it "an isolated word without etymology."
Latin costa developed a secondary sense in Medieval Latin of "the shore," via notion of the "side" of the land, as well as "side of a hill," and this passed into Romanic (Italian costa "coast, side," Spanish cuesta "slope," costa "coast"), but only in the Germanic languages that borrowed it is it fully specialized in this sense (Dutch kust, Swedish kust, German Küste, Danish kyst).
French also used this word for "hillside, slope," which led to the English verb meaning "a slide or sled down a snowy or icy hillside," first attested 1775 in American English. Expression the coast is clear (16c.) is an image of landing on a shore unguarded by enemies; to clear the coast (1520s) was to make it suitable for landing.
late 14c., "to skirt, to go around the sides, to go along the border" of something (as a ship does the coastline), from Anglo-French costien, from the French source of coast (n.).
The meaning "sled downhill," first attested 1834 in American English, is a separate borrowing or a new development from the noun. In bicycle-riding, "descend a hill with the feet off the pedals," from 1879. Of motor vehicles, "to move without thrust from the engine," from 1896; figurative use, of persons, "not to exert oneself," by 1934. Related: Coasted; coasting.
"Coasting" consists in throwing the legs up over the handles and allowing the bicycle to rush of its own impetus down hill. It can only be done with safety where the road is perfectly smooth, hard, and free from obstructions; but, under such conditions, bicycle coasting affords one of the most glorious and exhilarating of sensations, and, next to ballooning, its motion most nearly resembles the flight of a bird. [Harper's Weekly, Dec. 20, 1879]
The reckless coasting down the long hills on the route was scarcely more defensible. Speeds of 25 to 30 miles an hour were reached in some instances. The common road is not the proper place for such exhibitions, especially in populous centres. The risk is altogether too great, both for occupants of the vehicle and for other frequenters of the highway. [account of an automobile race on the streets of New York in "The Horseless Age," June 1896]