Recall the Taylor series Sum(x^n/n!). The same series can be used to define e^z for a complex number z=a+bi.
Use the Taylor series to show that exp(iy) = cos(y) + i sin(y) for any real number y. To do this substitute iy into the series and compute several terms. Look for patterns.
The solution demonstrates how to utilize Taylor series to obtain the famous Euler's identity