The range of a2sec2x+b2cosec2x where a,b∈R+ and x∈0,π2

Since a2sec2x+b2cosec2x=a2(1+cot2x)+b2(1+tan2x)=a2+b2+a2cot2x+b2tan2x ⇒A.M≥G.M a2cot2x+b2tan2x2≥    a2cot2 x    b2   tan2  x                  ≥a2+b2+2ab (using A.M-G.M inequality)=(a+b)2