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

Explanation

Putting, `x^(2)=cos2theta`, we get
`tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2))))=tan^(-1)((sqrt(1+cos2theta)+sqrt(1-cos2theta))/(sqrt(1+cos2theta)-sqrt(1-cos2theta)))`
`tan^(-1) (costheta+sintheta)/(costheta-sintheta)=tan^(-1)(1+tantheta)/(1-tantheta)`. [dividing num. and denom. by `cos theta`]
`=tan^(-1){tan(pi/4+theta)}=(pi/4+theta)`
`=pi/4+1/2cos^(-1)x^(2)`
`therefore tan^(-1) ((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2))))=pi/4+1/2cos^(-1)x^(2)`.