The solution set of inequality `(cot^(-1)x)(tan^(-1)x)+(2-pi/2),cot^(-1)x-3tan^(-1)x-3(2-pi/2)>0` is `(a , b),` then the value of `cot^(-1)a+cot^(-1)b` is____

Answer

5

Explanation

`(cot^(-1) x) (tan^(-1)x) + (2 -(pi)/(2)) cot^(-1) x - 3 tan^(-1) x 3 (2 - (pi)/(2)) gt 0`
`rArr cot^(-1) x (tan^(-1) x -(pi)/(2)) + 2 cot^(-1) x - 6 - 3 (tan^(-1) x -(pi)/(2)) gt 0`
`rArr -(cot^(-1) x)^(2) + 5 cot^(-1) x - 6 gt 0`
`rArr (cot^(-1) x -3) (2 - cot^(-1) x) gt 0`
`rArr (cot^(-1) x -3) (cot^(-1) x -2) lt 0`
`rArr 2 lt cot^(-1) x lt3`
`rArr cot 3 lt x lt cot2` [as `cot^(-1) x` is a decreasing function]
`rArr` Hence, `x in (cot3, cot2)`
`rArr cot^(-1) a + cot^(-1) b = cot^(-1) (cot 3) + cot^(-1) (cot 2) = 5`