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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{(1-costheta)/sintheta}`
`=tan^-1{(2sin^2 theta/2)/(2sin theta/2cos theta/2)}`
`=tan6-1{tan(theta/2)}`
`=theta/2`
`=(tan^-1x)/2`
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