Coefficient of x6 in eix+e-ix2 is

eix+e-ix2 =(cos x+i sin x)+(cos x-i sin x)2 =cos x =1-x22+x44-x66+....x6 coef =-16

Coefficient of x6 in eix+e-ix2 is

eix+e-ix2 =(cos x+i sin x)+(cos x-i sin x)2 =cos x =1-x22+x44-x66+....x6 coef =-16

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Coefficient of $x^{6}$ in $\frac{e^{ix} + e^{- ix}}{2}$ is

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$\frac{1}{6}$

$- \frac{1}{6}$

$- \frac{1}{6!}$

$\frac{1}{6!}$

answer is D.

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Detailed Solution

$\begin{array}{rcl} \frac{e^{ix} + e^{- ix}}{2} \\ = & \frac{(\cos x + i \sin x) + (\cos x - i \sin x)}{2} \\ = & \cos x \\ = & 1 - \frac{x^{2}}{2} + \frac{x^{4}}{4} - \frac{x^{6}}{6} + . . . . \end{array}$
$\begin{array}{rcl} x^{6} coef & = & \frac{- 1}{6} \end{array}$