Find the minimum value of `(sec^(-1)x)^2+("c o s e c"^(-1))^2`

Explanation

Let `I = (sec^(-1) x)^(2) + (cosec^(-1) x)^(2)`
`= (sec^(-1) x + cosec^(-1) x)^(2) - 2 sec^(-1) x cosec^(-1) x`
`= (pi^(2))/(4) - 2 sec^(-1) x ((pi)/(2) - sec^(-1) x)`
`= (pi^(2))/(4) + 2 (sec^(-1) x)^(2) - pi sec^(-1) x`
`= (pi^(2))/(4) + 2 [(sec^(-1) x)^(2) - 2(pi)/(4) sec^(-1) x + ((pi)/(4))^(2)] - (pi^(2))/(8)`
`= 2 (sec^(-1) x - (pi)/(4))^(2) + (pi^(2))/(8) rArr I ge (pi^(2))/(8)`