Prove that`tan^(-1)((sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)-sqrt(1-x)))=pi/4-1/2cos^(-1),-1/(sqrt(2))lt=xlt=1`

Explanation

let x `=cos theta implies cos^(-1)x`
`tan^(-1)((sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x)))`
`tan^(-1)((sqrt(1+cos theta)-sqrt(1-costheta))/(sqrt(1+cos theta)+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)((cos""(theta)/(2)-sin""(theta)/(2))/(cos""(theta)/(2)+sin""(theta)/(2)))=tan^(-1)((1-tan""(theta)/(2))/(1+tan""(theta)/(2)))`
`= tan^(-1)tan((pi)/(4)-(theta)/(2))=(pi)/(4)-(theta)/(2)`
`=(pi)/(4)-(1)/(2) cos^(-1)x=`RHS Hence Proved.