once reciprocals are taken of both sides; Eq. (2.379) may then be rewritten as
(2.380)
given the rule of composition of powers – where combination with Eq. (2.369) yields
together with Eq. (2.373) upon replacement of θ by nθ. Since Eq. (2.378) may be rewritten as
as per Eqs. (2.369) and (2.372), one concludes that
following ordered addition of Eqs. (2.381) and (2.382), together with cancelation of symmetrical terms afterward; by the same token, ordered subtraction of Eq. (2.381) from Eq. (2.382) generates
In view of Eq. (2.375), one may calculate the power of a cosine via
and Eq. (2.377) similarly supports
after retrieving Newton’s binomial as per Eq. (2.236), it is possible to reformulate Eq. (2.385) to
(2.387)
where the powers of z and of its reciprocal may be lumped to yield
If the exponent of the cosine function is an even integer, say, 2n, then Eq. (2.388) can be redone to
after replacement of n by 2n as upper limit, and concomitant replacement of i by 2i as counting variable of the summation – with subsequent splitting of the said summation, so as to make the median term appear explicitly. At this stage, it is convenient to revisit Eq. (2.240) and realize that
(2.390)
following straightforward algebraic manipulation; in other words,
– i.e. the row entries of Pascal’s triangle are symmetrical relative to its median (see Table 2.1). On the other hand, one may introduce a new counting variable satisfying
so that the second summation in Eq. (2.389) can be algebraically converted to
In view of Eq. (2.391), one may reformulate Eq. (2.393) to
(2.394)
– which, in turn, supports conversion of Eq. (2.389) to
(2.395)
upon lumping of summations for sharing the same lower and upper boundaries, coupled with factoring out of
along with definition of a composite power. Equation (2.383) may finally be invoked to rewrite Eq. (2.396) as
after
