Edited sentence: And apply the binomial theorem.Due to symmetry, the sine term drops out and finally yields the curious identity.However, before integrating out terms, the exponents need to be shifted by  to ensure that the desired coefficient  is not annihilated.Now integrating both sides over  annihilates all terms except the desired binomial coefficient.Since it forms an orthogonal basis, it has the following property.The binomial coefficient can also be derived through the complex exponential.The idea is to use orthogonality to isolate a specific binomial coefficient.This equates the average of cosines over the interval  to the ratio of  to the sum of all combinations.This gives.To begin, expand cosine in terms of the complex exponential.To rid of the complex terms, apply Euler's formula on the left-hand side