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JEE Main202330 Jan 2023Evening ShiftMathematicsInverse Trigonometric FunctionsActual

Let a 1 = 1 , a 2 , a 3 , a 4 , … . . be consecutive natural numbers. Then tan - 1 1 1 + a 1 a 2 + tan - 1 1 1 + a 2 a 3 + … . . + tan - 1 1 1 + a 2021 a 2022 is equal to

Options

  1. Aπ 4 - cot - 1 ( 2022 )
  2. Bcot - 1 ( 2022 ) - π 4
  3. Ctan - 1 ( 2022 ) - π 4
  4. Dπ 4 - tan - 1 ( 2022 )

Correct answer

A. π 4 - cot - 1 ( 2022 )

Step-by-step solution

Given, tan - 1 1 1 + a 1 a 2 + tan - 1 1 1 + a 2 a 3 . . . . . . . . . . . . . . . . tan - 1 1 1 + a 2021 a 2022 = tan - 1 a 2 - a 1 1 + a 1 a 2 + tan - 1 a 3 - a 2 1 + a 2 a 3 . . . . . . . . . . . . . . . . tan - 1 a 2022 - a 2021 1 + a 2021 a 2022 = tan - 1 a 2 - tan - 1 a 1 + tan - 1 a 3 - tan - 1 a 2 . . . . . . . . . . + tan - 1 a 2022 - tan - 1 a 2021 using the formula    tan - 1 x - tan - 1 y = tan - 1 x - y 1 + x y = tan - 1 a 2022 - tan - 1 a 1 = tan - 1 a 2022 - tan - 1 1   as&#1

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