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Question
What is the simplest form of `tan^-1 sqrt(1 - x^2 - 1)/x, x ≠ 0`
Options
`2 tan^-1x`
`1/2`
`3 tan^-1 x`
`1/3`
MCQ
Solution
`1/2`
Explanation:
Given `tan^-1 sqrt(1 + x^2 - 1)/x`
Put x = tan θ ⇒ θ = tan–1x
∴ `tan sqrt(1 + x^2 - 2)/x = tan^-1 (sqrt(1 - tan^2 theta - 1)/tan theta)`
= `tan^-1 ((sec theta - 1)/ tan theta)`
= `tan^-1 ((1 - cos theta)/sin theta)`
`tan^-1 [(2 sin^2 theta/2)/(2sin theta/2 cos theta/2)] = tan^-1[tan theta/2] = theta/2 = 1/2 tan^-1x`
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