Find the range of `y=(cot^(-1)x)(cot^(-1)(-x))dot`

Explanation

`y = (cot^(-1) x) (cot^(-1) (-x))`
`= cot^(-1) (x) (pi - cot^(-1) (x))`
Now, `cot^(-1) (x) and (pi - cot^(-1) (x)) gt 0`
Using A.M. `ge` G.M., we get
`(cot^(-1) x + (pi - cot^(-1) x))/(2) ge sqrt((cot^(-1) x) (pi - cot^(-1) x))`
`rArr 0 lt cot^(-1) (x) (pi - cot^(-1) (x))`
`le ((cot^(-1) x + (pi - cot^(-1) x))/(2))^(2) = (pi^(2))/(4)`
`rArr 0 lt y le (pi^(2))/(4)`