The words waist, waste sound the same but have different meanings and spellings. Why do waist, waste sound the same even though they are completely different words?
The answer is simple: waist, waste are homophones of the English language.
The part of the human trunk between the bottom of the rib cage and the pelvis.
The narrow part of the abdomen of an insect.
The part of a garment that encircles the waist of the body.
The upper part of a garment, extending from the shoulders to the waistline, especially the bodice of a woman's dress.
To use, consume, spend, or expend thoughtlessly or carelessly.
To cause to lose energy, strength, or vigor; exhaust, tire, or enfeeble: Disease wasted his body.
To fail to take advantage of or use for profit; lose: waste an opportunity.
To destroy completely.
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.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").