The solution set of the inequality `tan^(-1)x+sin^(-1)x ge (pi)/(2)` is

Options

A. `[-1,1]`
B. `[sqrt((sqrt(5)-1)/(4)),1]`
C. `[sqrt((sqrt(5)-1)/(2)),1]`
D. `[(sqrt(5)-1)/(2),1]`

Answer

C

Explanation

`tan^(-1)x + sin^(-1)x ge (pi)/(2), x in [-1,1]`
`rArr tan^(-1)x ge (pi)/(2) -sin^(-1)x`
`rArr tan^(-1)x ge cos^(-1)x`
`rArr x ge "tan"^(-1)(sqrt(1-x^(2)))/(x)`
`rArr x ge (sqrt(1-x^(2)))/(x)`
`x^(4)ge 1-x^(2)`
`rArr x^(4)+x^(2)-1ge 0`
`rArr x in [sqrt((sqrt(5)-1)/(2)),1]`