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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NTA Abhyas 2020 · Mathematics · Inverse Trigonometric Functions

The value of x for which ( ^ -1 (1 + x)) = ( ^ -1 x ) is

The value of xx for which sin⁡(cot⁡−1(1+x))=cos⁡(tan⁡−1x)\sin \left(\cot^{-1} \left(1 + x\right)\right) = \cos ( \tan^{-1} x ) is

Options

  1. A12\dfrac{1}{2}
  2. B11
  3. C00
  4. D−12- \dfrac{1}{2}

Correct answer

D. −12- \dfrac{1}{2}

Step-by-step solution

Let, cot⁡−1(x+1)=θ\cot^{-1} \left(x + 1\right) = \theta and tan⁡−1x=ϕ\tan^{-1} x = ϕ

Now, given equation becomes sinθ=cosϕsin\theta = cosϕ

⇒11+(x+1)2=11+x2⇒ \dfrac{1}{\sqrt{1 + {\left(x + 1\right)}^{2}}} = \dfrac{1}{\sqrt{1 + x^{2}}}

⇒(x+1)2+1=x2+1⇒x=−12⇒ {\left(x + 1\right)}^{2} + 1 = x^{2} + 1 ⇒ x = - \dfrac{1}{2}
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