Find the set of value of x for which the equation `cos^(-1) x + cos^(-1) ((x)/(2) + (1)/(2) sqrt(3 -3x^(2))) = (pi)/(3)` holds goods

Answer

`x in [(1)/(2), 1]`

Explanation

`cos^(-1) x + cos^(-1) [(x)/(2) + (1)/(2) sqrt(3 -3x^(2))] = (pi)/(3)`
`rArr cos^(-1) x + cos^(-1) [(1)/(2) x + (sqrt3)/(2) sqrt(1 -x^(2))] = (pi)/(3)`
`rArr cos^(-1) x + cos^(-1).(1)/(2) - cos^(-1) x = (pi)/(3)`
Above holds when `cos^(-1).(1)/(2) ge cos^(-1) x`
`:. x ge (1)/(2)`
`:. x in [(1)/(2), 1]`