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