How have the courts interpreted the Miranda waiver?

Defendant makes three claims related to his statements to police: (1) the initial statements he made to the police following his arrest were involuntary because the police threatened to kill him and his intoxication made him susceptible to that threat; (2) these initial statements were obtained in violation of Miranda v. Arizona (1966) 384 U.S. 436, 86 S.Ct. 1602, 16 L.Ed.2d 694; and (3) because his initial statements were obtained in violation of Miranda, his subsequent statements, obtained after a valid Miranda waiver, were tainted.