Provide an appropriate response.
-Use the fact that $\tan x = x + \frac { x ^ { 3 } } { 3 } + \frac { 2 x ^ { 5 } } { 15 } + \frac { 17 x ^ { 7 } } { 315 } + \ldots$ for $| x |

Question

Provide an appropriate response.
-Use the fact that $\tan x = x + \frac { x ^ { 3 } } { 3 } + \frac { 2 x ^ { 5 } } { 15 } + \frac { 17 x ^ { 7 } } { 315 } + \ldots$ for $| x | < \frac { \pi } { 2 }$ to find the first four terms of the series for ln(cos x).
A) $- \left( 1 + x ^ { 2 } + \frac { 2 x ^ { 4 } } { 3 } + \frac { 17 x ^ { 6 } } { 45 } + \ldots \right)$
B) $1 + x ^ { 2 } + \frac { 2 x ^ { 4 } } { 3 } + \frac { 17 x ^ { 6 } } { 45 } + \ldots$
C) $\frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 6 } } { 45 } + \frac { 17 x ^ { 8 } } { 2520 } + \ldots$
D) $- \left( \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 12 } + \frac { x ^ { 6 } } { 45 } + \frac { 17 x ^ { 8 } } { 2520 } + \ldots \right)$

Quizplus · Accepted Answer

Using the identity $\ln (\cos x) = - \ln (\sec x)$ and the fact that $\sec x = \frac{1}{\cos x}$, we have:
\begin{align*}
\ln (\cos x) &= - \ln (\sec x) \
&= - \ln \left(\frac{1}{\cos x}ight) \
&= - \ln \left( 1 + (\cos x - 1)ight)  \
&= -\left( \frac{(\cos x - 1)}{1} + \frac{(\cos x - 1)^2}{2} + \frac{2(\cos x - 1)^3}{3} + \frac{17(\cos x - 1)^4}{45} + \dots ight) \quad 	ext{(from the given formula for }	an x	ext{)}
\end{align*}
Now, we just need to substitute $\cos x$ back in for the variable $x$ and simplify. The first four terms are:
\begin{align*}
-\left( \frac{(\cos x - 1)}{1} + \frac{(\cos x - 1)^2}{2} + \frac{2(\cos x - 1)^3}{3} + \frac{17(\cos x - 1)^4}{45}ight) = -\left(\frac{1}{2}x^2 + \frac{1}{12}x^4 + \frac{1}{45}x^6 + \frac{17}{2520}x^8 + \dots ight)
\end{align*}

Provide an Appropriate Response tan⁡x=x+x33+2x515+17x7315+…\tan x = x + \frac { x ^ { 3 } } { 3 } + \frac { 2 x ^ { 5 } } { 15 } + \frac { 17 x ^ { 7 } } { 315 } + \ldotstanx=x+3x3​+152x5​+31517x7​+…

Provide an Appropriate Response $\tan x = x + \frac { x ^ { 3 } } { 3 } + \frac { 2 x ^ { 5 } } { 15 } + \frac { 17 x ^ { 7 } } { 315 } + \ldots$