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Question
Write the following in the simplest form:
`tan^-1{(sqrt(1+x^2)+1)/x},x !=0`
Solution
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`
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