Question

If `|cos^-1((1-x^2)/(1+x^2))| < pi/3,` then x belongs to the interval
A. `[-1//sqrt(3),1//sqrt(3)]`
B. `(-1//sqrt(3),1//sqrt(3))`
C. `(0,1//sqrt(3))`
D. none of these

Answer 1

We have
`|cos^(-1)((1-x^(2))/(1+x^(2)))|lt(pi)/(3)` ltrbgt `rarr 1/2 lt (1-x^(2))/(1+x^(2)) lt (pi)/(3)`
`rarr 1+x^(2)lt2-2x^(2)lt2+2x^(2)`
`rarr 1+x^(2)lt2-2x^(2) and 2-2x^(2)lt2+2x^(2)`
`rarr 3x^(2)-1lt0and 4x^(2)gt0 rarr -(1)sqrt(3)ltxlt(1)/sqrt(3)`