Find `x` satisfying `[tan^(-1)x]+[cos^(-1)x]=2,` where `[]` represents the greatest integer function.

Explanation

`0 lt cot^(-1) x lt pi and -pi//2 lt tan^(-1) x lt pi//2`
`rArr [cot^(-1) x] in {0, 1, 2, 3} and [tan^(-1) x] in {-2, -1, 0, 1}`
For `[tan^(-1) x] + cot^(-1) x] = 2`, following cases are possible
Cose(i): `[cot^(-1)x] = tan^(-1) x] = 1`
`rArr 1 le cot^(-1) x lt 2 and 1 le tan^(-1) x lt pi//2`
`rArr x in (cot 2, cot 1] and x in [tan 1, oo)`
`:. x in phi " as " cot 1 lt tan 1`
Case (ii): `[cot^(-1) x] = 2, [tan^(-1) x] = 0`
`rArr 2 le cot^(-1) x lt 3 and 0 le tan^(-1) x lt 1`
`rArr x in (cot 3, cot 2] and x in [0, tan 1)`
`:. x in phi " as " cot 2 lt 0`
So no solution
Case (iii) : `[cot^(-1) x] = 3, [tan^(-1) x] = -1`
`rArr 3 le cot^(-1) x lt pi and - 1 le tan^(-1) x lt 0`
`rArr x in (-oo, cot 3] and x in [- tan 1, 0)`
`:. x in phi cot 3 lt - tan 1`
Therefore, no such value of x exist