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.
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The solution demonstrates how to utilize Taylor series to obtain the famous Euler's identity.