(2.383) may again be invoked to transform Eq. (2.411) to
(2.412)
while the denominator was rewritten as a product of composite powers – where ι2 = −1 can be used to generate
(2.413)
cancelation of (−1)n between numerator and denominator, and factoring out of 2 in the former unfold
since (−1)i−n coincides with (−1)n−i and x is more generally used as argument than angle θ (as long as rad is employed as units).
In the case of an odd exponent, Eq. (2.407) may be rephrased as
based on change of n to 2n + 1, and likewise of i to 2i + 1, complemented by splitting of the summation in two halves and rearrangement of exponents wherever appropriate; recalling Eq. (2.399), one obtains
as alternative form for the second summation in Eq. (2.415), where condensation of terms alike meanwhile took place. In view of Eq. (2.391) pertaining to the symmetry of binomial coefficients, one may proceed to
(2.417)
as new version of Eq. (2.416) – which may be inserted in Eq. (2.415) to generate
where (−1)2n = (−12)n = 1n = 1 was taken into account after factoring out, (−1)−2j was rewritten as (−1)(−1)−2j−1 = −(−1)−(2j + 1), and
along with rewriting of exponent 2n + 1 − 2i as 2(n − i) + 1 and 22n+1 as twice 22n, as well as lumping of powers of −1 between numerator and denominator. After retrieving Eq. (2.384) and replacing n by 2(n − i) + 1, one may reformulate Eq. (2.419) to
(2.420)
in view of (−1)i−n = 1/(−1)i−n = (−1)n−i and 22n = (22)n, which may instead look like
as more usual form – using x (expressed in rad) as independent variable rather than θ; Eqs. (2.414) and (2.421) accordingly permit calculation of an (integer) power of sine of any argument as a linear combination of cosines or sines of multiples of said argument.
The converse problem of expressing sines and cosines of nθ in terms of powers of sin θ and cos θ may also be solved via de Moivre’s theorem; one should accordingly retrieve Eq. (2.369), and expand its left‐hand side via Newton’s binomial formula as
– where advantage was meanwhile taken of ι2 = −1, ι3 = −ι, ι4 = 1, and Eq. (2.236). Following inspection of the forms of the terms in the right‐hand side of Eq. (2.422), one realizes that ι may be factored out to get
(2.423)
a more condensed notation is, however, possible according to
Equation (2.424) may be rewritten as
in the case of an even multiple of θ, materialized via replacement by 2n; and alternatively
when said multiple is odd, i.e. consubstantiated in 2n + 1. Note that no need exists here to change also the form of the counting variable, because no upper limit for the summation was (deliberately) provided in Eq.
