Solve the following equation: `sin^(-1)x+sin^(-1)(1-x)=cos^(-1)x`

Explanation

`sin^(-1) x + sin^(-1) (1-x) = cos^(-1)x `
`rArr " "sin[xsqrt(1-(1-x)^(2)) +(1-x)sqrt(1-x^(2))]= sin ^(-1)sqrt(1-x^(2))`
`rArr xsqrt(2x-x^(2)) + sqrt(1-x)- xsqrt(1-x^(2)) = sqrt(1-x^(2))`
`rArr " "xsqrt(2x-x^(2)) + xsqrt(1-x^(2)) = 0`
`rArr" "x[sqrt(2x-x^(2)) -sqrt(1-x^(2))]=0`
`rArr x = 0 or sqrt(2x-x^(2))-sqrt(1-x^(2))=0` Now `sqrt(2x-x^(2))-sqrt(1-x^(2)) = 0`
`rArr" "sqrt(2x -x^(2)) = sqrt(1-x^(2))`
`rArr" "2x -x^(2) = 1- x^(2)`
`rArr " "2x =1`
`rArr " " x=1//2`
` therefore " "x=0 or x = 1//2`