\begin{array}{l}
 e^{ix} = 1 + \frac{{ix}}{{1!}} - \frac{{x^2 }}{{2!}} - \frac{{ix^3 }}{{3!}} + \frac{{x^4 }}{{4!}} + \frac{{ix^5 }}{{5!}} - \frac{{x^6 }}{{6!}} - \frac{{ix^7 }}{{7!}} + \frac{{x^8 }}{{8!}} + ... = \\ 
 = \left( {1 - \frac{{x^2 }}{{2!}} + \frac{{x^4 }}{{4!}} - \frac{{x^6 }}{{6!}} + \frac{{x^8 }}{{8!}} - ...} \right) + i\left( {\frac{x}{{1!}} - \frac{{x^3 }}{{3!}} + \frac{{x^5 }}{{5!}} - \frac{{x^7 }}{{7!}} + ...} \right) = \cos x + i\sin x \\ 
 \end{array}