Question

Write the following in the simplest form:

`tan^-1{(sqrt(1+x^2)-1)/x},x !=0`

Answer 1

Let x = tan θ

Now,

`tan^-1{(sqrt(1+x^2)-1)/x}=tan^-1{(sqrt(1+tan^2theta)-1)/tantheta}`

`=tan^-1  {(sqrt(sec^2theta)-1)/tantheta}`

`=tan^-1{(sectheta-1)/tantheta}`

`=tan^-1{(1-costheta)/sintheta}`

`=tan^-1{(2sin^2  theta/2)/(2sin  theta/2cos  theta/2)}`

`=tan6-1{tan(theta/2)}`

`=theta/2`

`=(tan^-1x)/2`