Question-136475

Question Number 136475 by EDWIN88 last updated on 22/Mar/21

Commented by EDWIN88 last updated on 22/Mar/21

Use the fact that \csc x = \frac{1}{x} + \frac{x}{6} + \frac{{7 x}^{3}}{360} + \frac{{31 x}^{5}}{15120} + \dots for ∣ x ∣ < π to

find the first four terms of the series for

2 \csc (2 x) [for ∣ x ∣ < \frac{π}{2} and thus for \ln (\tan x) .

(a) \ln ∣ x ∣ + \frac{x^{2}}{12} + \frac{{7 x}^{4}}{1440} + \frac{{31 x}^{6}}{90720} + \dots

(b) \ln ∣ x ∣ - \frac{x^{2}}{3} + \frac{{7 x}^{4}}{90} - \frac{{62 x}^{6}}{2835} + \dots

(c) \ln ∣ x ∣ + \frac{x^{2}}{3} + \frac{{7 x}^{4}}{90} + \frac{{62 x}^{6}}{2835} + \dots

(d) \ln ∣ x ∣ - \frac{x^{2}}{12} + \frac{{7 x}^{4}}{1440} - \frac{{31 x}^{6}}{90720} + \dots

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