We can write the cosine part as: [2.2] Where k is the term of the expansion of the cosines.
Because (i·sin x)2k=(i2·sin2 x)k, we can write: [2.3]
Because, i2=−1, and −sin2 x=(cos2 x−1), [2.4]
Noting that the binomial expansion of (cos2x−1)k is: [2.5] Where m is the term of this expansion, then [2.6]
the cosines together. We can bring them into the second sum, because
the k does not change, is effectively constant for any given k [2.7] This is the direct expression for cos(nx).
While m is the term of the expansion 2.5, it is also the term of the expression for cos(nx), when we write it in the order of low powers of cosine to higher, and simplify. This is not true for the sines. ■
If we wish to find the mth term of cos(nx), we note that the power of
cos(nx) required is n-2m, so that m is a constant for that term, so we
can write: [2.8] In other words, there is only one m value for each k, so there is no sum over m (one value only).
The term itself is: [2.9] We begin with k=m, because unless k≥m, then the binomial coefficients will disappear.
We can use 2.9 to compute a term of cos(nx) directly. ■