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NTA Abhyas JEE Main2020MathematicsInverse Trigonometric FunctionsPractice

If cot - 1 ⁡ x - x 2 2 + x 3 4 - . … … + tan - 1 ⁡ x 2 - x 4 2 + x 6 4 - . … … = π 2 for 0 < x < 2 , then x is equal to

Options

  1. A0 only
  2. B1 only
  3. C0 , 1 both
  4. DNone of these

Correct answer

B. 1 only

Step-by-step solution

Since tan - 1 ⁡ x + cot - 1 ⁡ x = π 2 for x ∈ R . ∴ x - x 2 2 + x 3 4 - . … . . = x 2 - x 4 2 + x 6 4 - . … . . ⇒ x 1 + x 2 = x 2 1 + x 2 2 0 < x < 2 ⇒ x 2 + x = x 2 2 + x 2 ⇒ 2 x + x 3 = 2 x 2 + x 3 ⇒ x 2 = x ⇒ x = 0 , 1 But x ≠ 0 , ∴ x = 1 .

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