`cos^(-1){1/2x^(2)+sqrt(1-x^(2))sqrt(1-(x^(2))/(4))}=cos^(-1)((x)/(2))-cos^(-1)x` holds for what values of x?

Explanation

`cos^(-1) ((1)/(2) x^(2) + sqrt(1 -x^(2)) sqrt(1 - (x^(2))/(4))) = cos^(-1) (x.(x)/(2) + sqrt(1 -x^(2)) sqrt(1 - ((x)/(2))^(2)))`
for `cos^(-1) ((1)/(2) x^(2) + sqrt(1 - x^(2)) sqrt(1 - (x^(2))/(4))) = cos^(-1) (x)/(2) - cos^(-1) x`
L.H.S. `gt 0`, hence R.H.S. `gt 0`
`rArr cos^(-1).(x)/(2) - cos^(-1) gt 0 " or " cos^(-1).(x)/(2) gt cos^(-1) x`
Since `cos^(-1) x` is a decreasing function, we get
`(x)/(2) le x rArr (x)/(2) ge 0 rArr x ge 0 rArr x in [0,1]`