Prove that `cos^(-1) {sqrt((1 + x)/(2))} = (cos^(-1) x)/(2) , -1 lt x lt 1`

Explanation

Let `x = cos theta`, where `theta in [0, pi]`. Then,
`cos^(-1) {sqrt((1 + x)/(2)) = cos^(-1) {sqrt((1 + cos theta)/(2))}`
`= cos^(-1) {sqrt((2 cos^(2).(theta)/(2))/(2))}`
`= cos^(-1) (cos.(theta)/(2)) ( :' cos.(theta)/(2) gt 0)`
`= (theta)/(2) = (cos^(-1) x)/(2)`