I'm 99% certain I've found this, but oh my goodness it's an obscure one. The story is Mutiny by Malcolm Hulke and it has appeared only in two little known anthologies Galactic Adventures and Purnell's Book of Adventures in Space (the latter has been mentioned in Who wrote "The Monsters" short story published by Purnell?).
The only trace I can find of the story in the Google Books page on Galactic Adventures, though that page is only partially complete. Only fragmentary extracts of the book text are available from Google Books, but the few extracts available make it fairly clear this is the story in question. For example here we find:
... computer only followed instructions that began with the captain's voice saying please. That's why he was always polite to the computer. And we just erased all his voice-prints from the tape. He's fooled us.