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AP EAMCET202119 Aug 2021Morning ShiftMathematicsInverse Trigonometric FunctionsActual

If tan - 1 1 1 + 1 . 2 + tan - 1 1 1 + 2 . 3 + ⋯ + tan - 1 1 1 + n ( n + 1 ) = tan - 1 [ x ] , then x =

Options

  1. A1 n + 1
  2. Bn n + 1
  3. C1 n + 2
  4. Dn n + 2

Correct answer

D. n n + 2

Step-by-step solution

We have, tan - 1 1 1 + 1 . 2 = tan - 1 2 - 1 1 + 1 . 2 Now, using the formula tan - 1 x - tan - 1 y = tan - 1 x - y 1 + x y ,   x y > - 1 We get, tan - 1 1 1 + 1 . 2 = tan - 1 2 - tan - 1 1 Similarly, tan - 1 1 1 + 2 . 3 = tan - 1 3 - tan - 1 2 and tan - 1 1 1 + n ( n + 1 ) = tan - 1 n + 1 - tan - 1 n ⇒ tan - 1 1 1 + 1 . 2 + tan - 1 1 1 + 2 . 3 + ⋯ + tan - 1 1 1 + n ( n + 1 ) = tan - 1 2 - tan - 1 1 + tan - 1 3 - tan - 1 2 + ⋯ + tan - 1 n + 1 - tan - 1 n = tan - 1 n + 1 - tan - 1 1 = tan

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