Complete solution set of `(cot^(-1)x)+2(tan^(-1)x)=0,` where `[]` denotes the greatest integer function, is equal to `(0,cot1)` (b) `(0,tan1)` `(tan1,oo)` (d) `(cot1,tan1)`

Options

A. `(0, cot 1)`
B. `(0, tan 1)`
C. `(tan 1, oo)`
D. `(cot 1, tan 1)`

Answer

D

Explanation

`[cot^(-1) x] + 2 [tan^(-1) x] = 0`
`rArr [cot^(-1) x] = 0, [tan^(-1) x] = 0`
or `[cot^(-1) x] = 2, [tan^(-1) x] = -1`
Now `[cot^(-1) x] = 0 rArr x in (cot 1, oo)`
`[tan^(-1) x] = 0 rArr x in (0, tan 1)`
Hence, `x in (cot 2, tan 1)`
Therefore,
for `[cot^(-1) x] = [tan^(-1) x] = 0, x in (cot 1, tan 1)`
`[cot^(-1) x] = 2 rArr x in (cot 3, cot 2)`
`[tan^(-1) x] = -1 rArr x in [-tan 1, 0)`
Hence, `x in [-tan 1, cot 2]`