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we're in our quest to give ourselves a little bit of a mathematical underpinning of definite integrals of various combinations of trig functions so it'll be hopefully straightforward for us to actually find the coefficients our Fourier coefficients which we're going to do a few videos from now and we've already started going down this path we've established that the definite integral from 0 to 2pi of sine of M T DT is equal to 0 that the cosine the definite integral of cosine M T DT is equal to 0 for any nonzero integer M actually we can generalize that a little bit for sine of M T it could be for any M actually and I if you don't believe me I encourage you let me write this for any for any integer M this top integral is going to be 0 and this second integral for any four nonzero non zero integer integer M and you can see if you had zero in this second case it would be cosine of 0 T so this would just evaluate to 1 so you just be integrating the value 1 from 0 to 2pi and so that's going to have a non zero value so with those two out of the way let's let's go a little bit deeper get a little bit more foundations so I'm now I now want to establish that the definite integral from 0 to 2pi of sine of M T times cosine of of cosine of n at T and T DT that this equals 0 for any any integers integers m and n and they could even be the same M they don't have to necessarily be different but they could be different how do we do this well let's just rewrite let's just rewrite this part right over here leveraging some trig identities and if it's completely unfamiliar to you I encourage you to review your trig identities on Khan Academy so this is the same thing as a definite integral from 0 to 2pi this sine of M T times cosine NT we can rewrite it at using the product to sum formulas so that is let me use different color here so this thing right over here that I've underlined in magenta I'm squaring off in magenta that can be re-written as 1/2 times sine of M plus n T sine of M plus n T plus sine sine of M minus n M minus n t and then let me just close that with a DT DT now if we were to just rewrite this using some of our integral properties we could rewrite it as so this part over here we could and let's assume we distribute the 1/2 so we're going to distribute the 1/2 and use some of our integral properties and so what are we going to get so this part roughly right over here we could rewrite as 1/2 times the definite integral from 0 to 2 pi of sine of M plus n T DT and then this part once you distribute the 1/2 and you use some integration properties this could be plus 1/2 times the definite integral from 0 to 2pi of sine of M minus n m- n T DT now what are each of these things going to be equal to well isn't this right over here isn't that just some integer if I have to take the sum of two arbitrary integers that's going to be some integer so that's going to be some integer and this 2 is going to be some integer right over here and we've already established that the definite integral of sine of some integer times T DT is 0 so by this first thing that we already showed this is going to be equal to 0 and that's going to be equal to 0 doesn't matter that you're multiplying by 1/2 1/2 times 0 0 1/2 times 0 is 0 this whole thing is going to evaluate to 0 so there you go we've Reuven that as well