Question

Sove `[cot^(-1) x] + [cos^(-1) x] =0`, where `[.]` denotes the greatest integer function

Answer 1

We have `[cot^(-1) x] + [cos^(-1) x] =0`
Since `cos^(-1)x, cot^(-1) x ge 0`, we must have
`[cot^(-1) x] = [cos^(-1) x] =0`
Now, `[cot^(-1) x] = 0`
`rArr 0 lt cot^(-1) x lt 1`
`rArr x in (cot 1, oo)`...(1)
Also, `[cos^(-1) x] = 0` ltrbgt `rArr 0 le cos^(-1) x lt1`
`rArr x in (cos 1, 1]`...(2)
From (1) and (2) `x in (cot 1,10`