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JEE Main202117 Mar 2021Morning ShiftMathematicsInverse Trigonometric FunctionsActual

If cot - 1 ( α ) = cot - 1 2 + cot - 1 8 + cot - 1 18 + cot - 1 32 + … . upto 100 terms, then α is:

Options

  1. A1 . 01
  2. B1 . 00
  3. C1 . 02
  4. D1 . 03

Correct answer

A. 1 . 01

Step-by-step solution

Given cot - 1 ( α ) = cot - 1 ( 2 ) + cot - 1 ( 8 ) + cot - 1 ( 18 ) + … . = ∑ n = 1 100 tan - 1 2 4 n 2 = ∑ n = 1 100 tan - 1 ( 2 n + 1 ) - ( 2 n - 1 ) 1 + ( 2 n + 1 ) ( 2 n - 1 ) = ∑ n = 1 100 tan - 1 2 n + 1 - tan - 1 2 n - 1 = tan - 1 201 - tan - 1 1 = tan - 1 200 202 ∴ cot - 1 ( α ) = cot - 1 202 200 α = 1 . 01

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