Cofficient of `x^6` in `(e^(ix)+e^(-ix))/2` is

Options

A. `1/(6!)`
B. `1/6`
C. `-1/6`
D. `-1/(6!)`

Answer

To find the coefficient of \( x^6 \) in the expression \( \frac{e^{ix} + e^{-ix}}{2} \), we can follow these steps:

### Step 1: Simplify the Expression
We start with the expression:
\[ \frac{e^{ix} + e^{-ix}}{2} \]
Using the Euler's formula, we know that:
\[ e^{ix} + e^{-ix} = 2\cos(x) \]
Thus, we can rewrite our expression as:
\[ \frac{e^{ix} + e^{-ix}}{2} = \cos(x) \]

### Step 2: Use the Taylor Series Expansion for Cosine
The Taylor series expansion for \( \cos(x) \) around \( x = 0 \) is given by:
\[ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \ldots \]
This series continues indefinitely.

### Step 3: Identify the Coefficient of \( x^6 \)
From the expansion, we can see that the term containing \( x^6 \) is:
\[ -\frac{x^6}{6!} \]
Thus, the coefficient of \( x^6 \) is:
\[ -\frac{1}{6!} \]

### Step 4: Conclusion
Therefore, the coefficient of \( x^6 \) in the expansion of \( \frac{e^{ix} + e^{-ix}}{2} \) is:
\[ -\frac{1}{720} \]
since \( 6! = 720 \).

### Final Answer
The coefficient of \( x^6 \) is \( -\frac{1}{720} \).

---