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