The solution of the inequality `(log)_(1/2)(sin)_(1/2)sin^(-1)x >(log)_(1//2)cos^(-1)x` is `x in [(0,1)/(sqrt(2))]` (b) `x in [1/(sqrt(2)),1]` `x in ((0,1)/(sqrt(2)))` (d) none of these

Options

A. `x in [0, (pi)/(sqrt2)]`
B. `x in ((1)/(sqrt2), 1]`
C. `x in (0, (1)/(sqrt2))`
D. none of these

Answer

C

Explanation

`"log"_(1//2) sin^(-1) x gt "log"_(1//2) cos^(-1) x`
`rArr cos^(-1) x gt sin^(-1) x, 0 lt x lt 1`
`rArr cos^(-1) x gt (pi)/(2) - cos^(-1) x, 0 lt x lt 1`
`rArr cos^(-1) x gt (pi)//(4), 0 lt x lt 1`
`rArr 0 lt x lt (1)/(sqrt2)`