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If -1 lt x lt 0, then cos^(-1) x is equa...

If `-1 lt x lt 0`, then `cos^(-1) x` is equal to

A

`sec^(-1).(1)/(x)`

B

`pi - sin^(-1) sqrt(1 -x^(2))`

C

`pi + tan^(-1).(sqrt(1 -x^(2)))/(x)`

D

`cot^(-1).(x)/(sqrt(1 -x^(2)))`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
`cos^(-1) x = sec^(-1).(1)(x) " for all " x in [-1, 0)`
Leet `x = -y`
`:. Cos^(-1) x = cos^(-1) (-y) = pi - cos^(-1) y`
`= pi - sin^(-1) sqrt(1 -y^(2))`...(i)
`=pi - tan^(-1).(sqrt(1 - y^(2)))/(y)` ..(ii)
From (i), `cos^(-1) = pi - sin^(-1) sqrt(1 - y^(2)) = pi - sin^(-1) sqrt(1 -x^(2))`
From (ii), `cos^(-1) x = pi - tan^(-1).(sqrt(1 -y^(2)))/(y)`
`= pi - tan^(-1).(sqrt(1 -x^(2)))/(-x)`
`= pi + tan^(-1).(sqrt(1 -x^(2)))/(x)`
`= pi + cot^(-1).(x)/(sqrt(1 -x^(2))) -pi " " ("as " x lt 0)`
`= cot^(-1).(x)/(sqrt(1 -x^(2)))`
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