The values of x which satisfy `18(sin^(-1)x)^(2)-9pi sin^(-1)x +pi^(2)lt 0` and `18(tan^(-1)x)^(2)-9pi tan^(-1)x + pi^(2)lt 0` simultaneously are

Options

A. `((sqrt(3))/(3),(sqrt(3))/(2))`
B. `((sqrt(3))/(2),1)`
C. `((1)/(3),(sqrt(3))/(2))`
D. `((1)/(sqrt(3)),sqrt(3))`

Answer

A

Explanation

`(sin^(-1)x)^(2)-(pi)/(2)sin^(-1)x+(pi^(2))/(18)lt 0`
`rArr (sin^(-1)x-pi//3)(sin^(-1)x-pi//6)lt 0`
`rArr (pi)/(6)lt sin^(-1)x lt(pi)/(3)`
`rArr (1)/(2)lt x lt (sqrt(3))/(2)` ….(1)
Similarly
`(pi)/(6)lt tan^(-1)x lt (pi)/(3)`
`rArr (1)/(sqrt(3))lt x lt sqrt(3)` ....(2)
From (1) and (2), we get `(1)/(sqrt(3))lt x lt (sqrt(3))/(2)`