Powers of Sine and Cosine as Multiple-Angle Sums
Summary: §261–§263. Any integer power of or can be written as a finite linear combination of sines (or cosines) of multiple angles , with binomial-coefficient weights. This is the inverse of §234: instead of expanding in powers of , Euler expands in . The derivation uses only the four product-to-sum lemmas and induction.
Sources: chapter14 (§261–§263)
Last updated: 2026-05-11
The lemma (§262)
The whole reduction rests on the four product-to-sum identities:
2\sin a\sin z &= \cos(a-z) - \cos(a+z) \\ 2\cos a\sin z &= \sin(a+z) - \sin(a-z) \\ 2\sin a\cos z &= \sin(a+z) + \sin(a-z) \\ 2\cos a\cos z &= \cos(a-z) + \cos(a+z) \end{aligned}$$ (source: chapter14, §262). Each multiplication of an existing $\sin kz$ or $\cos kz$ by another $\sin z$ or $\cos z$ produces two terms whose arguments differ by $\pm z$. Iterating gives a sum supported on $\{0, \pm z, \pm 2z, \ldots, \pm nz\}$ collapsed by parity. ## Powers of sine (§262) Multiplying the previous power's expansion by $\sin z$ via the lemma and collecting: $$\begin{aligned} \sin z &= \sin z \\ 2(\sin z)^2 &= 1 - \cos 2z \\ 4(\sin z)^3 &= 3\sin z - \sin 3z \\ 8(\sin z)^4 &= 3 - 4\cos 2z + \cos 4z \\ 16(\sin z)^5 &= 10\sin z - 5\sin 3z + \sin 5z \\ 32(\sin z)^6 &= 10 - 15\cos 2z + 6\cos 4z - \cos 6z \\ 64(\sin z)^7 &= 35\sin z - 21\sin 3z + 7\sin 5z - \sin 7z \\ 128(\sin z)^8 &= 35 - 56\cos 2z + 28\cos 4z - 8\cos 6z + \cos 8z \\ 256(\sin z)^9 &= 126\sin z - 84\sin 3z + 36\sin 5z - 9\sin 7z + \sin 9z \end{aligned}$$ (source: chapter14, §262). The leading factor on the left is $2^{n-1}$. The non-constant coefficients are the **binomial coefficients $\binom{n}{k}$ from the row indexed by $n$**, signs alternating, and using $\sin$ for odd $n$ and $\cos$ for even $n$. The constant term in even $n$ is the central binomial coefficient $\binom{n}{n/2}$ of the previous (also even-indexed) row, equivalently the binomial coefficient at the centre of row $n$. ## Powers of cosine (§263) The same procedure with $\cos z$ in place of $\sin z$ — but using $2\cos a\cos z = \cos(a-z) + \cos(a+z)$ instead of $2\sin a\sin z = \cos(a-z) - \cos(a+z)$ — yields: $$\begin{aligned} \cos z &= \cos z \\ 2(\cos z)^2 &= 1 + \cos 2z \\ 4(\cos z)^3 &= 3\cos z + \cos 3z \\ 8(\cos z)^4 &= 3 + 4\cos 2z + \cos 4z \\ 16(\cos z)^5 &= 10\cos z + 5\cos 3z + \cos 5z \\ 32(\cos z)^6 &= 10 + 15\cos 2z + 6\cos 4z + \cos 6z \\ 64(\cos z)^7 &= 35\cos z + 21\cos 3z + 7\cos 5z + \cos 7z \end{aligned}$$ (source: chapter14, §263). All signs are positive; only $\cos kz$ appears (no sines), and the coefficients again follow the binomial law. ## Modern reading These are the standard *power-reduction formulas*. The cleanest derivation uses [[eulers-formula|Euler's formula]] $\cos z = (e^{iz} + e^{-iz})/2$ and $\sin z = (e^{iz} - e^{-iz})/(2i)$ together with the binomial theorem: $$(2\cos z)^n = (e^{iz} + e^{-iz})^n = \sum_{k=0}^{n}\binom{n}{k}e^{i(n - 2k)z},$$ which immediately produces Euler's tables on grouping conjugate pairs. Euler does not write the derivation this way in §261–§263 — he stays with the real product-to-sum lemma — but the result and its binomial-coefficient pattern are the same. The pattern matters for integration: $\int(\cos z)^n\,dz$ becomes a sum of $\int\cos(kz)\,dz$, each of which is elementary. This is precisely how Euler and his successors evaluated such integrals before the substitution $u = \tan(z/2)$ became standard. ## Earlier use in Chapter 13 §222 The odd-power half of the §262 table appears earlier in the book, in [[chapter-13-on-recurrent-series|Chapter 13]] §222, where Euler invokes (without proof at that point) $$4(\sin\phi)^3 = 3\sin\phi - \sin 3\phi,\qquad 16(\sin\phi)^5 = 10\sin\phi - 5\sin 3\phi + \sin 5\phi,\qquad 64(\sin\phi)^7 = \cdots$$ to convert the complex-derived [[general-term-of-recurrent-series|general term]] of $A/(1 - 2pz\cos\phi + p^2 z^2)^k$ back to real form for $k = 2, 3, 4$ (the denominators $(\sin\phi)^3, (\sin\phi)^5, (\sin\phi)^7$ that appear in those formulas are cleared using exactly these identities). The Chapter 14 derivation supplies the proof Euler postponed. ## Inverse relationship to Chapter 14's main thread The chapter's main movement (§234–§256) expands $\sin nz$ as a polynomial in $\sin z$ — the [[multiple-angle-polynomials|Chebyshev]] direction. §261–§263 is the inverse: it expands $(\sin z)^n$ as a sum of $\sin kz$ (or $\cos kz$). Together the two directions form an invertible change of basis between $\{(\sin z)^k : k = 0, 1, \ldots, n\}$ and $\{1, \sin z, \cos z, \sin 2z, \cos 2z, \ldots\}$ on each finite-dimensional space of trig polynomials. ## Related pages - [[multiple-angle-polynomials]] - [[trigonometric-addition-formulas]] - [[de-moivre-formula]] - [[eulers-formula]] - [[binomial-series]] - [[sine-and-cosine]] - [[general-term-of-recurrent-series]] - [[chapter-13-on-recurrent-series]] - [[chapter-14-on-the-multiplication-and-division-of-angles]]