证明y=tanx在（-π/2,π/2）上为单调递增函数要按定义严格证明

Question

证明y=tanx在（-π/2,π/2）上为单调递增函数
要按定义严格证明

Answer 1

设 -π/2则 tan(X1)-tan(X2)
=sin(X1)/cos(X1)-sin(X2)/cos(X2)
=[sin(X1)cos(X2)-sin(X2)cos(X1)]/cos(X1)cos(X2)
=sin(X1-X2)/cos(X1)cos(X2)
则 (X1-X2)∈(-π,0) 即sin(X1-X2)<0
且 cos(X1)cos(X2)>0
即tan(X1)-tan(X2)<0
由题设-π/2于是y=tanx在（-π/2,π/2）上为单调递增函数