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NTA Abhyas JEE Main2020MathematicsInverse Trigonometric FunctionsPractice

The value of the expression cot -1 1 2 + cot -1 9 2 + cot -1 2 5 2 + cot -1 4 9 2 + ....... upto n terms is

Options

  1. Atan - 1 ⁡ 2 n
  2. Btan - 1 ⁡ 2 n - 1
  3. Ctan - 1 ⁡ n
  4. Dtan - 1 ⁡ 2 n - tan - 1 ⁡ 1

Correct answer

A. tan - 1 ⁡ 2 n

Step-by-step solution

Given expression = tan -1  2 + tan -1  2 9 + tan -1  2 2 5 + tan -1  2 4 9 + ........ General term = 2 2 n - 1 2 = 2 4 n 2 - 4 n + 1 = 2 1 + 4 n n - 1 = 2 n - 2 n - 2 1 + 2 n 2 n - 2 T n = tan - 1 ⁡ 2 n - tan - 1 ⁡ 2 n - 2 ∴ Sum of the series = tan - 1 ⁡ 2 - tan - 1 ⁡ 0 + tan - 1 ⁡ 4 - tan - 1 ⁡ 2 + tan - 1 ⁡ 6 - tan - 1 ⁡ 4 + … tan - 1 ⁡ 2 n - tan - 1 ⁡ 2 n - 2 = tan - 1 ⁡ 2 n - tan - 1 ⁡ 0 = tan - 1 ⁡ 2 n

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