 An important question to ask is, why study complex analysis? Well, probably because your program requires it, but more generally, for many problems, the complex version of the problem is actually easier than the real version. We'll illustrate this with trigonometric identities. For example, we can prove the cosine of x plus y is cosine x cosine y minus sine x sine y using geometry. A lot of geometry. X and Y are real angles, and their sum is between zero and pi halves. But what if they're not? As a warm-up, let's prove our basic trigonometric identity, sine squared plus cosine squared equal to one, using the exponential forms. So those exponential forms are, and let's find sine squared and cosine squared in terms of those exponential forms. So from the exponential form of sine theta, we find. And from the exponential form of cosine theta, we find. So when we add sine squared plus cosine squared, we get. Now the Pythagorean identity isn't too difficult to prove, but what about a more complicated identity? So let's prove our cosine sum formula. So first, we can write down cosine of x plus y in exponential form. And remember, part of the value of proof is that it reviews what we know. We should know the exponential form of sine and cosine. And we also know the rules of exponents, so we can simplify the e to power i times x plus y as. And similarly for e to power negative i times x plus y. Now we don't know if this is going to be useful, but we might as well see where it takes us. Now remember the last line of a proof is what you've proven. We want to get cosine x cosine y minus sine x sine y, so let's write that as our last line and hope we can get there. And again, we have the exponential forms for sine and cosine. So we can find cosine x cosine y and sine x sine y. And equals means replaceable no matter which direction you're going. So we can put those as the preceding line. And let's do a little algebra. And now we have this gap between where we ended up and where we want to go. So we stare at this for a moment and try and figure out what we can do. And you might notice that if we multiply numerator and denominator by 2, we can bridge the gap. And since we started with cosine x plus y and ended with cosine x cosine y minus sine x sine y, that proves our statement. Now once you've seen a proof like this that uses the exponential forms, you'll probably agree that we don't really need the suggestion to use the exponential forms. It's what we want to do anyway. Because remember, complex is easier. So let's talk calculus.