If `-1lexle0` then `sin^(-1)x` equals

Options

A. `pi-sin^(-1)sqrt(1-x^(2))`
B. `tan^(-1)((x)/sqrt(1-x^(2)))`
C. `-cot^(-1)(sqrt(1-x^(2))/(x))`
D. none of these

Explanation

Let `sin^(-1)x=theta` Then `x =sin theta`
Now `-1 lt x lt 0 rarr-(pi)/(2)ltthetalt0`
`pi-sin^(-1)sqrt(1-x^(2))`
`=pi-sin^(-1)(cos theta)`
`As-(pi)/(2)ltthetalt0`
`therefore 0ltpi-thetalt(pi)/(2)rarr 0ltpi-sin^(-1)sqrt(1-x^(2))lt(pi)/(2)`
`therefore sin^(-1)x ne pi-sin^(-1)sqrt(1-x6(2))`
So option (a) is not correct
we have
`tan^(-1)(x)/sqrt(1-x^(2))`
`=tan^(-1)(sin htheta)/(sqrt(1-sin^(2)theta)=tan^(-1)(tan theta)=theta =sin^(-1)x`
Thus option (b) is correct
We have
`-cot^(-1)sqrt(1-x^(2))/(x)=cos^(-1)(cos theta)//(sin theta)=-cos^(-1)(cot theta)`
`=cosT^(-1)(cos(-theta))`
`-theta`
`-sin^(-1)x`
Thus option (c ) is also not correct