已知|x|≤1,|y|≤1,求证|x+y/1+xy|≤1.

题目

已知|x|≤1,|y|≤1,求证|x+y/1+xy|≤1.
已知|x|≤1,|y|≤1,求证|x+y/1+xy|≤1 "| |"代表绝对值

答案

|x+y/1+xy|=|x+y/1+xy-1+1|
=|(-1-xy+x+y)/(1+xy)+1|
=|-(1-x)(1-y)/(1+xy)+1|
|x|≤1,|y|≤1
1-x>=0
1-y>=0
1+xy>=0
-(1-x)(1-y)/(1+xy)