If `sin^(-1)(1-x) sin^(-1)x=(pi)/(2)` then x equal

Options

A. `-1/2`
B. 1
C. 0
D. `1/2`

Explanation

We have, `sin^(-1)(1-x)-2sin^(-1)x=pi/2`
`sin^(-1)(1-x)=pi/2+2sin^(-1)x`
`1-x=sin((pi)/(2)+sin ^(-1)x)`
`1-x+cos (2sin^(-1)x)`
`1-x=cos(2sin^(-1)x)`
`1-x=cos[cos^(-1)(1-2x^(2))]`
`1-x=1-2x^(2)`
`2x^(2)-x=0`
` x(2x-1)=0`
`therefore x=0or x=1/2`
For `x=1/2`
`sin^(-1)(1-x)-2sin^(-1)((1)/(2))-2sin^(-1)((1)/(2))=-sin^(-1)((1)/(2))=(-pi)/(6)`
So, `x=1/2` is not the solution of the given equation. For `x=0`
`sin^(-1)(1-x)-2sin^(-1)(1)-2sin^(-1)(0)`
`=pi/2-0=pi/2`
Hence, the correct anaswer from the given alternative is (c).