Question

Write the following in the simplest form:

`tan^-1{sqrt(1+x^2)-x},x in R`

Answer 1

Let x = cot θ

Now,

`tan^-1{sqrt(1+x^2)-x}=tan^-1{sqrt(1+cot^2theta)-cottheta}`

`=tan^-1{cosectheta-cottheta}`

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

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

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

`=theta/2`

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