If `|cos^(-1)((1-x^2)/(1+x^2))| < pi/3,t h e n` (a) `x in [-1/3,1/(sqrt(3))]` (b) `x in [-1/(sqrt(3)),1/(sqrt(3))]` `x in [0,1/(sqrt(3))]` (d) none of these

Options

A. `x in [-(1)/(3), (1)/(sqrt3)]`
B. `x in (-(1)/(sqrt3), (1)/(sqrt3))`
C. `x in (0, (1)/(sqrt3))`
D. none of these

Answer

B

Explanation

We have
`|cos^(-1) ((1 -x^(2))/(1 + x^(2)))| lt (pi)/(3)`
`rArr -(pi)/(3) lt cos^(-1) ((1 -x^(2))/(1 + x^(2))) lt (pi)/(3)`
`rArr 0 le cos^(-1).(1 -x^(2))/(1 + x^(2)) lt (pi)/(3)`
`rArr (1)/(2) lt (1 - x^(2))/(1 + x^(2)) le 1`
`rArr 1 + x^(2) lt 2 (1 -x^(2)) le 2 (1 + x^(2))`
`rArr 0 le x^(2) lt (1)/(3)`
`rArr -(1)/(sqrt3) lt x lt (1)/(sqrt3)`