Prove that `tan^(-1) {(x)/(a + sqrt(a^(2) - x^(2)))} = (1)/(2) sin^(-1).(x)/(a), -a lt x lt a`

Explanation

Let `x = a sin theta, -a lt x lt a`. Then,
`-a lt a sin theta lt a`
or `-1 lt sin theta lt 1 rArr theta in (-(pi)/(2), (pi)/(2))`
`rArr tan^(-1) {(x)/(a+ sqrt(a^(2) - x^(2)))} = tan^(-1) {(a sin theta)/(a + sqrt(a^(2) - a^(2) sin^(2) theta))}`
`= tan^(-1) {(sin theta)/(1 + cos theta)}`
`= tan^(-1) {(2 sin (theta//2) cos (theta//2))/(2 cos^(2) (theta//2))}`
`= tan^(-1) {tan.(theta)/(2)}`
`= (theta)/(2) = (1)/(2) sin^(-1). (x)/(a)`