trig-infinite-products

Infinite Products for Tangent, Cotangent, Secant, Cosecant

Summary: §186–§187: dividing the §184 linear-factor products for $\sin (m\pi /2n)$ by the corresponding ones for $\cos (m\pi /2n)$ yields infinite product representations for $\tan (m\pi /2n)$, $\cot (m\pi /2n)$, $\sec (m\pi /2n)$, $\csc (m\pi /2n)$. Replacing $m$ with another integer $k$ in the §184 formulas gives products for ratios $\sin (m\pi /2n)/\sin (k\pi /2n)$ and similar — once one trig value is known, all the others at the same denominator $2n$ follow without further computation.

Sources: chapter11

Last updated: 2026-05-01

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Tangent and cotangent

Dividing the first sine expression by the first cosine expression in §184:

$\tan \frac{m\pi }{2n}=\frac{m\pi /2n\cdot (2n-m)/2n\cdot (2n+m)/2n\cdots }{(n-m)/n\cdot (n+m)/n\cdot (3n-m)/3n\cdot (3n+m)/3n\cdots }$

Pairing numerator and denominator factors of similar size and simplifying:

$\boxed{\tan \frac{m\pi }{2n}=\frac{m}{n-m}\cdot \frac{2n-m}{n+m}\cdot \frac{2n+m}{3n-m}\cdot \frac{4n-m}{3n+m}\cdot \frac{4n+m}{5n-m}\cdots }$

(source: chapter11, §186). The cotangent is the reciprocal:

$\cot \frac{m\pi }{2n}=\frac{n-m}{m}\cdot \frac{n+m}{2n-m}\cdot \frac{3n-m}{2n+m}\cdot \frac{3n+m}{4n-m}\cdot \frac{5n-m}{4n+m}\cdots$

Secant and cosecant

The secant is $1/\cos$, but with the §184 cosine in the denominator form, the simplest expression is

$\sec \frac{m\pi }{2n}=\frac{n}{n-m}\cdot \frac{n}{n+m}\cdot \frac{3n}{3n-m}\cdot \frac{3n}{3n+m}\cdot \frac{5n}{5n-m}\cdot \frac{5n}{5n+m}\cdots$

Likewise, $\csc (m\pi /2n)=1/\sin (m\pi /2n)$:

$\csc \frac{m\pi }{2n}=\frac{n}{m}\cdot \frac{2n}{2n-m}\cdot \frac{2n}{2n+m}\cdot \frac{4n}{4n-m}\cdot \frac{4n}{4n+m}\cdots$

(source: chapter11, §186). Each is a straightforward rearrangement of one of the four §184 products.

Variants from the second pair

Using the second §184 expression (the one obtained via the co-function identity) in the quotient gives a different-looking formula for the same value. For tangent:

$\tan \frac{m\pi }{2n}=\frac{\pi }{2}\cdot \frac{m}{n-m}\cdot \frac{1}{2}\cdot \frac{2n-m}{n+m}\cdot \frac{3}{2}\cdot \frac{2n+m}{3n-m}\cdot \frac{3}{4}\cdot \frac{4n-m}{3n+m}\cdots$

(source: chapter11, §186). The redundancy mirrors the §185 Wallis phenomenon: two formulas for the same value differ by a Wallis-style factor that telescopes to $\pi /2$.

Ratios — §187

Replace $m$ by another integer $k$ in the §184 sine formula: the ratio

$\frac{\sin (m\pi /2n)}{\sin (k\pi /2n)}$

is an infinite product whose factors come in pairs from the corresponding terms of the two products. For instance:

$\frac{\sin (m\pi /2n)}{\sin (k\pi /2n)}=\frac{m}{k}\cdot \frac{2n-m}{2n-k}\cdot \frac{2n+m}{2n+k}\cdot \frac{4n-m}{4n-k}\cdot \frac{4n+m}{4n+k}\cdots$

(source: chapter11, §187). Analogous formulas hold for $\sin /\cos$, $\cos /\cos$, etc. Euler’s remark: “if we take $k\pi /2n$ as an angle whose sine and cosine are known, by means of the above formulas, we can find the sine and cosine of any other angle $m\pi /2n$” (source: chapter11, §187). One trig table entry generates all the others at the same denominator.

Why two expressions per function?

Because the §184 linear-factor split gives two product formulas for each of $\sin (m\pi /2n)$ and $\cos (m\pi /2n)$ — one direct, one via the co-function — the four functions $\tan ,\cot ,\sec ,\csc$ all inherit two expressions each. Comparing the two routinely yields the Wallis-style identity $\pi /2=(2/1)(2/3)(4/3)(4/5)\cdots$ as a “calibration constant” between them.

Computational use

These products are not the practical route to numerical values of the trig functions: like Wallis, they converge geometrically in $1/k^2$ and need many factors per digit. Their value in chapter 11 is structural, not computational:

• The redundancy generates the Wallis product (§185).
• The ratio formulas (§187) reduce the number of independent table entries needed.
• The §188–§198 log trick — taking $\log$, expanding via the §118 series, transposing the resulting double sum — converts even slowly-convergent products like these into fast-convergent series. See log-sine-via-products.

For direct numerical computation of $\tan ,\cot$, the better route is the §181 partial-fraction expansion, which Euler picks up in §197–§198 of this same chapter.

Related pages

• linear-factors-of-sine-cosine
• sine-infinite-product
• cosine-infinite-product
• wallis-product
• log-sine-via-products
• cotangent-partial-fraction
• trigonometric-addition-formulas
• chapter-11-on-other-infinite-expressions-for-arcs-and-sines