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Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

MathematicsInverse Trigonometric FunctionsSummation of seriesMedium2 minQB

MathematicsMediumsingle choice

Passage / Comprehension

For positive adjacent values $z_{0}, z_{1}, \dots, z_{n}, tan^{- 1} ((z_{r} - z_{r - 1}) / (1 + z_{r} z_{r - 1})) = tan^{- 1} z_{r} - tan^{- 1} z_{r - 1}$ , so the sum telescopes.

Find $csc^{- 1} (5) + csc^{- 1} (65) + csc^{- 1} (325) + \dots$

Options:

Answer:

Solution:

The $r$ th term is $csc^{- 1} ((1 + 4 r^{4})) = cot^{- 1} (2 r^{2}) = tan^{- 1} [2/ (4 r^{2})]$ . This equals $tan^{- 1} (2 r + 1) - tan^{- 1} (2 r - 1)$ , so the sum tends to $π /2 - tan^{- 1} 1 = π /4$ .

Stream:JEESubject:MathematicsTopic:Inverse Trigonometric FunctionsSubtopic:Summation of series

⏱ 2mℹ️ Source: QB

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