Prove that `tan^(-1)\ (sqrt(1+x^2)-1)/x=1/2tan^(- 1)x.`

Explanation

Putting, `x=tantheta`, we get
`tan^(-1)(sqrt(1+x^(2))-1)/(x)=tan^(-1)(sqrt(1+tan^(2)theta-1)/(tantheta))`
`=tan^(-1)(sectheta-1)/(tantheta)=tan^(-1)(1-costheta)/(sintheta)`
`=tan^(-1){(2sin^(2)theta/2)/(2sintheta/2costheta/2)}=tan^(-1)(tantheta/2)`
`=theta/2=1/2tan^(-1)x`.
`therefore tan^(-1)((sqrt(1+x^(2))-1)/(x)=1/2tan^(-1)x`.