The solution set of the equation
`cos^(-1)x-sin^(-1)x=sin^(-1)(1-x)` is

Options

A. `[-1,1]`
B. `[0,1//2]`
C. `[-1,0]`
D. none of these

Explanation

We observe that the LHS of the given equation is defined for `-1lexle1` where RHS is defined fro `-1le(1-x)le1 i.e 0 le x le2` so the given equation holds for `x in [0,1]`
`cos^(-1)x-sin^(-1)x=sin^(-1)(1-x)`
`rarr (pi)/(2)-2sin^(-2)x=sin^(-1)(1-x)`
`rarr (pi)/(2)-sin^(-1)(1-x)=2sin^(-1)`
`rarr cos^(-1)(1-x)=cos^(-1)(1-2x^(2))`
`rarr 1-x=1-2x^(2) rarr 2x^(2) -x=0 rarr xc =0 1/2`