If `(sin^(-1) x)^(2) - (cos^(-1) x)^(2) = a pi^(2)` then find the range of a

Answer

A

Explanation

`a pi^(2) = (sin^(-1) x)^(2) - (cos^(-1) x)^(2)`
`rArr a pi^(2) = (sin^(-1) x - cos^(-1) x) (sin^(-1) x + cos^(-1) x)`
`rArr a pi^(2) = (sin^(-1) x - cos^(-1) x) (pi//2)`
`rArr 2 a pi = (sin^(-1) x cos^(-1) x)`
`rArr 2a pi = (2 sin^(-1) x - pi//2)`
Now, `-pi le 2 sin^(-1) x le pi`
`rArr -3pi//2 le 2 sin^(-1) x - pi//2 le pi//2`
`:. -3 pi//2 le 2a pi le pi//2`
or `-3//4 le a le 1//4`