hotmail login just loops back to the sign in page

try a private window first. if it signs in fine there, it's cookies or an extension in your normal browser, so clear cookies for the microsoft and live sites and turn off any privacy/ad blockers for a minute. still looping in private mode? check the PC's date and time are set automatically. a clock that's off by a few minutes breaks the sign-in token and you get bounced exactly like that.