Find the sum of all possible values of x satisfying arc `cos((2)/(pi) arc cos x)=arc sin((2)/(pi) arc sinx)`.

Explanation

1
arc `cos((2)/(pi) arc cos x)=arc sin ((2)/(pi) arc sin x) `
`cos^(-1)((2)/(pi) ((pi)/(2)-sin^(-1)x))=sin^(-1)((2)/(pi) sin^(-1)x)`
`cos^(-1)(1-(2)/(pi) sin^(-1)x)=sin^(-1)((2)/(pi) sin^(-1)x)`
Let `(2)/(pi) sin^(-1)x=alpha` were `alpha in [0,1]` think !
`implies cos^(-1)(1-alpha)=sin^(-1) alpha`
`implies sin^(-1) sqrt(2alpha-alpha^(2))=sin^(-1) alpha implies sqrt(2alpha-alpha^(2))=alphaimplies 2 alpha-alpha^(2)=alpha^(2)`
`implies 2 alpha=2alpha^(2)`
Hence `alpha` is either 0 or 1
If `alpha=0` then x=0
if `alpha=1` then x=1
hence sum of all possible value of x is 1