Question

Write the following in the simplest form:

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

Answer 1

Let x = cot θ

Now,

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

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

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

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

`=tan^-1{cot  theta/2}`

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

`=(pi/2-theta/2)`

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