Prove that `sin^(-1) {(sqrt(1 + x) + sqrt(1 - x))/(2)} = (pi)/(4) + (cos^(-1) x)/(2), 0 lt x lt 1`

Explanation

Let `x = cos theta`.
Since `0 lt x lt 1, theta in (0, (pi)/(2))`
`rArr sin^(-1) {(sqrt(1 + x) + sqrt(1 -x))/(2)}`
`= sin^(-1) {(sqrt(1 + cos theta) + sqrt(1 - cos theta))/(2)}`
`= sin^(-1) {(sqrt(2 cos^(2). (theta)/(2)) + sqrt(2 sin^(2). (theta)/(2)))/(2)}`
`= sin^(-1) {(cos.(theta)/(2) + sin.(theta)/(2))/(sqrt2)}`
`= sin^(-1) {sin((pi)/(4) + (theta)/(2))}`
`= (pi)/(4) + (theta)/(2) [ :' theta in (0, (pi)/(2)) rArr ((pi)/(4) + (theta)/(2)) in ((pi)/(4), (pi)/(2))]`
`= (pi)/(4) + (cos^(-1) x)/(2)`