Trig Multiple-Angle Partial Fractions

Summary: §237, §246–§256. By reading Vieta’s reciprocal-root formula off the multiple-angle polynomial, Euler expresses $n/\sin nz$, $n\cot nz$, $n\sec nz$, $n\csc nz$, and $n\cot nz$ (tangent version) each as a sum of $n$ terms involving the same function at equally-spaced shifted angles. These generalize the §181–§183 partial fractions of Chapter 10 from a fixed denominator to a sliding one.

Sources: chapter14 (§237, §246–§256)

Last updated: 2026-05-10

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Cosecant sum (§237)

From the reciprocal-root Vieta formula applied to the odd-$n$ polynomial in $x=\sin z$, the sum of reciprocals of all $n$ roots equals the ratio of the $(n-2)$-degree coefficient to the constant term. Euler writes this as:

$\frac{n}{\sin nz}=\frac{1}{\sin z}+\frac{1}{\sin (\frac{2\pi }{n}+z)}+\frac{1}{\sin (\frac{4\pi }{n}+z)}+\cdots$

where there are $n$ terms (source: chapter14, §237). In terms of cosecants:

$n\csc nz=\csc z+\csc (\frac{2\pi }{n}+z)+\cdots$

Example $n=3$ (§237): $\frac{3}{\sin 3z}=\frac{1}{\sin z}+\frac{1}{\sin (\frac{2\pi }{3}+z)}+\frac{1}{\sin (\frac{4\pi }{3}+z)}=\frac{1}{\sin z}+\frac{1}{\sin (\frac{\pi }{3}-z)}-\frac{1}{\sin (\frac{\pi }{3}+z)}$

The general formula for odd $n=2m+1$ (§237): $n\csc nz=\csc z+\csc (\frac{\pi }{n}-z)-\csc (\frac{\pi }{n}+z)-\csc (\frac{2\pi }{n}-z)+\csc (\frac{2\pi }{n}+z)+\cdots \pm \csc (\frac{m\pi }{n}+z)$

(source: chapter14, §248).

Cotangent sum (§237)

Dividing the cosecant formula by the matching product formula for $\sin nz$ and differentiating with respect to $z$ (implicitly), or equivalently by reading another Vieta coefficient, Euler also states:

$\frac{n\cos nz}{\sin nz}=\frac{\cos z}{\sin z}+\frac{\cos (\frac{2\pi }{n}+z)}{\sin (\frac{2\pi }{n}+z)}+\cdots$

i.e., $n\cot nz=\cot z+\cot (\frac{2\pi }{n}+z)+\cdots$ — a sum of $n$ cotangents (source: chapter14, §250).

The general formula for odd $n=2m+1$ (§251): $n\cot nz=\cot z-\cot (\frac{\pi }{n}-z)+\cot (\frac{\pi }{n}+z)-\cot (\frac{2\pi }{n}-z)+\cot (\frac{2\pi }{n}+z)-\cdots$

For even $n=2m$ (§256): $n\cot nz=-\tan z+\cot (\frac{\pi }{n}-z)-\cot (\frac{\pi }{n}+z)+\cot (\frac{2\pi }{n}-z)-\cdots +\cot (\frac{m\pi }{n}-z)$

(source: chapter14, §256). The alternating signs arise from $\cot v=-\cot (\pi -v)$.

Secant sum (§246)

From the cosine polynomial (§243), whose leading coefficient is 1 (the equation begins with 1), applying the same Vieta procedure:

$\frac{n}{\cos nz}=\frac{1}{\cos z}+\frac{1}{\cos (\frac{2\pi }{n}+z)}+\cdots$

For odd $n=2m+1$, the general formula (§246):

$n\sec nz=\sec (\frac{m}{n}\pi +z)+\sec (\frac{m}{n}\pi -z)-\sec (\frac{m-1}{n}\pi +z)-\cdots \pm \sec z$

(source: chapter14, §247). Euler works out examples for $n=1,3,5,7$ explicitly.

Tangent and cotangent via De Moivre (§249–§256)

Setting $t=\tan z$ and expanding $(1+ti{)}^n$ via the binomial theorem:

$\tan nz=\frac{nt-\left(\genfrac{}{}{0ex}{}{n}{3}\right)t^3+\left(\genfrac{}{}{0ex}{}{n}{5}\right)t^5-\cdots }{1-\left(\genfrac{}{}{0ex}{}{n}{2}\right)t^2+\left(\genfrac{}{}{0ex}{}{n}{4}\right)t^4-\cdots }$

(source: chapter14, §249). For odd $n$, the numerator has degree $n$ and the denominator has degree $n-1$. The $n$ roots of the numerator (when $\tan nz$ is set to a fixed value) are $\tan z,\tan (\frac{\pi }{n}+z),\tan (\frac{2\pi }{n}+z),\dots$

From the coefficient of the second term (§252, §255), the sum of all $n$ roots is $n\cot nz$ when the equation begins with the constant 1. For odd $n=2m+1$ (§253):

$n\tan nz=\tan z-\tan (\frac{\pi }{n}-z)+\tan (\frac{\pi }{n}+z)-\cdots +\tan (\frac{m\pi }{n}+z)$

For even $n=2m$ (§255–§256), comparing the highest-power equation with the factored form gives:

$n\cot nz=-\tan z+\tan (\frac{\pi }{n}-z)-\tan (\frac{\pi }{n}+z)+\cdots +\tan (\frac{m\pi }{n}-z)$

Product of tangents (§254, §257)

For odd $n$, the product of all $n$ tangent roots equals $\tan nz$ (source: chapter14, §254):

$\tan nz=\tan z\cdot \tan (\frac{\pi }{n}-z)\tan (\frac{\pi }{n}+z)\cdots \tan (\frac{m\pi }{n}-z)\tan (\frac{m\pi }{n}+z)$

For even $n$, the product of all $n$ tangent roots equals $1$ (§257):

$1=\tan z\cdot \tan (\frac{\pi }{n}-z)\tan (\frac{\pi }{n}+z)\cdots$

because the roots come in complementary pairs $\tan \alpha \cdot \tan (\pi /2-\alpha )=1$, so each pair has product $1$ (source: chapter14, §257).

Comparison with Chapter 10 (§181–§183)

The §181–§183 formulas express $\pi \cot (\pi \sqrt{a})$ and $\pi /\sin (\pi \sqrt{a})$ as series ${\sum }_{k}1/(k^2-a)$. The Chapter 14 formulas are the finite-$n$ analogue: $n\cot nz$ as a finite sum of $n$ cotangents. As $n\to \infty$ with $nz=\pi z_0$ fixed, each term $\cot (k\pi /n+z)$ contributes $\cot (k\pi /n)$, and the sum converges to the Chapter 10 Mittag-Leffler expansion. Chapter 14 thus provides the finite scaffolding from which Chapter 10’s infinite partial fractions emerge as a limit.

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