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

Answer

`cos sin 1 lt x lt cos cos 1`

Explanation

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)`