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