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

Explanation

Putting, `x=costheta`, we get
LHS `=tan^(-1){(sqrt(1+costheta)-sqrt(1-costheta))/(sqrt(1+costheta)+sqrt(1-costheta))}`
`=tan^(-1){(sqrt(2cos^(2)(theta/2))-sqrt(2sin^(2)(theta/2))/(sqrt(2cos^(2)theta/2))+sqrt(2sin^(2)theta/2))}=tan^(-1){(costheta/2-sintheta/2)/(costheta/2+sintheta/2)}`
`=tan^(-1){(1+tantheta/2)/(1+tantheta/2)}` [dividing num. and denom. by `costheta/2`].
`=tan^(-1){tan(pi/4-theta/2)}=(pi/4-theta/2)`
`=(pi/4-1/2cos^(-1)x)`=RHS.
Hence, LHS=RHS.