函數(shù)的定義域為：{x∈R|x≠kπ/2,k∈Z}
設(shè)t=sinx+cosx=√2sin(x+π/4),t∈[-√2,-1)∪(-1,1)∪(1,√2]
則tanx+cotx=1/(sinxcosx)=2/(t^2-1)
secx+cscx=(sinx+cosx)/(sinxcosx)=2t/(t^2-1)
所以y=t+2(t+1)/(t^2-1)=t-1+2/(t-1)+1
由對勾函數(shù)的單調(diào)性可求得：
y∈(-∞,1-2√2]∪[2+3√2,+∞)