Find the value of x for which `sin^(-1) (cos^(-1) x) lt 1 and cos^(-1) (cos^(-1) x) lt 1`

Find the value of x for which `sin^(-1) (cos^(-1) x) lt 1 and cos^(-1)

1.	Find the value of x for which `sin^(-1) (cos^(-1) x) lt 1 and cos^(-1) (cos^(-1) x) lt 1`
Answer» Correct Answer - `cos sin 1 lt x lt cos cos 1` We have `sin^(-1) (cos^(-1) x) lt 1` `rArr -(pi)/(2) le sin^(-1) (cos^(-1) x) lt 1` `rArr - 1 le cos^(-1) x lt sin 1` `rArr 0 le cos^(-1) x lt sin 1` `rArr cos (sin 1) lt x le 1`..(i) Also, `cos^(-1) (cos^(-1) x) lt 1` `rArr 0 le cos^(-1) (cos^(-1) x) lt 1` `rArr cos 1 lt cos^(-1) x le 1` `rArr cos 1 le x lt cos (cos 1)`....(ii) `rArr cos 1 lt cos (sin 1)` Hence, from (i) and (ii), we have `cos (sin 1) lt x lt cos (cos 1)`

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