Provide an appropriate response.
-Use the fact that $\cot x = \frac { 1 } { x } - \left( \frac { x } { 3 } + \frac { x ^ { 3 } } { 45 } + \frac { 2 x ^ { 5 } } { 945 } + \ldots \right)$ for $| x |

Question

Provide an appropriate response.
-Use the fact that $\cot x = \frac { 1 } { x } - \left( \frac { x } { 3 } + \frac { x ^ { 3 } } { 45 } + \frac { 2 x ^ { 5 } } { 945 } + \ldots \right)$ for $| x | < \pi$ to find the first four terms of the series for ln(sin $x$ ).
A) $- \left( \frac { 1 } { x ^ { 2 } } + \frac { 1 } { 3 } + \frac { x ^ { 2 } } { 15 } + \frac { 2 x ^ { 4 } } { 189 } + \ldots \right)$
B) $\frac { 1 } { x ^ { 2 } } + \frac { 1 } { 3 } + \frac { x ^ { 2 } } { 15 } + \frac { 2 x ^ { 4 } } { 189 } + \ldots$
C) $\ln | x | - \left( \frac { x ^ { 2 } } { 6 } + \frac { x ^ { 4 } } { 180 } + \frac { x ^ { 6 } } { 2835 } + \ldots \right)$
D) $- \ln | x | + \frac { x ^ { 2 } } { 6 } + \frac { x ^ { 4 } } { 180 } + \frac { x ^ { 6 } } { 2835 } + \ldots$

Quizplus · Accepted Answer

Given that $\cot x = \frac{1}{x} - \left( \frac{x}{3} + \frac{x^3}{45} + \frac{2x^5}{945} + \ldots \right)$, we can integrate both sides to find the series for $\ln(\sin x)$. Integrating $\cot x$ gives $\ln(\sin x)$, and integrating the series term by term gives $\ln |x| - \left( \frac{x^2}{6} + \frac{x^4}{180} + \frac{x^6}{2835} + \ldots \right)$.

Provide an Appropriate Response cot⁡x=1x−(x3+x345+2x5945+…)\cot x = \frac { 1 } { x } - \left( \frac { x } { 3 } + \frac { x ^ { 3 } } { 45 } + \frac { 2 x ^ { 5 } } { 945 } + \ldots \right)cotx=x1​−(3x​+45x3​+9452x5​+…)

Provide an Appropriate Response $\cot x = \frac { 1 } { x } - \left( \frac { x } { 3 } + \frac { x ^ { 3 } } { 45 } + \frac { 2 x ^ { 5 } } { 945 } + \ldots \right)$