Solve `(tan^(-1) x)^(2) + (cot^(-1) x)^(2) = (5pi^(2))/(8)`

Options

A. `[-(3)/(4), (1)/(4)]`
B. `[-(3)/(4), (3)/(4)]`
C. `[-1, 1]`
D. `[-1, (3)/(4)]`

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`