The words gait, gate sound the same but have different meanings and spellings. Why do gait, gate sound the same even though they are completely different words?
The answer is simple: gait, gate are homophones of the English language.
A particular way or manner of moving on foot: a person who ran with a clumsy, hobbling gait.
Any of the ways, such as a canter, trot, or walk, by which a horse can move by lifting the feet in different order or rhythm.
Rate or manner of proceeding: The project went forward at a steady gait.
To train (a horse) in a particular gait or gaits.
A structure that can be swung, drawn, or lowered to block an entrance or a passageway.
An opening in a wall or fence for entrance or exit.
The structure surrounding such an opening, such as the monumental or fortified entrance to a palace or walled city.
A means of access: the gate to riches.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.