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What is the simplest form of tan-1 1-x2-1x,x≠0 -

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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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