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

Explanation

`L.H.S. = tan^-1[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]`

`=tan^-1[(sqrt(1+x)(1-sqrt(1-x)/sqrt(1+x)))/(sqrt(1+x)(1+sqrt(1-x)/sqrt(1+x)))]`

`=tan^-1[(1-sqrt(1-x)/sqrt(1+x))/(1+sqrt(1-x)/sqrt(1+x))]`

As, `tan^-1x-tan^-1y = (x-y)/(1+xy)`

So, our expression becomes,

`=tan^-1(1)+tan^-1(sqrt(1-x)/sqrt(1+x))`

`=pi/4+1/2(2tan^-1(sqrt(1-x)/sqrt(1+x)))`

Also, `2tan^-1y = cos^-1((1-y^2)/(1+y^2))`

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