Inverse Cotangent

The function $\cot^{-1}x$, also denoted arccot($x$), where $\cot x$ is the Cotangent and the superscript $-1$ denotes an Inverse Function and not the multiplicative inverse. The Maclaurin Series is given by

$$\cot^{-1}x = {\textstyle{1\over 2}}\pi-x+{\textstyle{1\over ... ...^5+{\textstyle{1\over 7}}x^7-{\textstyle{1\over 9}}x^9+\ldots,$$ (1)

and Laurent Series by

$$\cot^{-1}x = x^{-1}-{\textstyle{1\over 3}}x^{-3}+{\textstyle... ...extstyle{1\over 7}}x^{-7}+{\textstyle{1\over 9}}x^{-9}+\ldots.$$ (2)

Euler derived the Infinite series

$$\cot^{-1} x=x\left[{{1\over x^2+1}+{2\over 3(x^2+1)^2}+{2\cdot 4\over 3\cdot 5(x^2+1)^3}+\ldots}\right]$$ (3)

(Wetherfield 1996).

The inverse cotangent satisfies

$$\cot^{-1}x=\pi-\cot^{-1}(-x)$$ (4)

for Positive and Negative $x$, and

$$\cot^{-1}={\textstyle{1\over 2}}\pi-\cot^{-1}\left({1\over x}\right)$$ (5)

for $x\geq 0$. The inverse cotangent is given in terms of other inverse trigonometric functions by

$$\cot^{-1}x = \cos^{-1}\left({x\over\sqrt{x^2+1}}\right)$$ (6)

$$= {\textstyle{1\over 2}}\pi-\sin^{-1}\left({x\over\sqrt{x^2+1}}\right)$$ (7)

$$= {\textstyle{1\over 2}}\pi+\tan^{-1}(-x)={\textstyle{1\over 2}}\pi-\tan^{-1}x$$ (8)

for Positive or Negative $x$, and

$$\cot^{-1}x = \csc^{-1}(\sqrt{x^2+1}\,)$$ (9)

$$= \sec^{-1}\left({\sqrt{x^2+1}\over x}\right)$$ (10)

$$= \sin^{-1}\left({1\over\sqrt{x^2+1}}\right)$$ (11)

$$= \tan^{-1}\left({1\over x}\right)$$ (12)

for $x\geq 0$.

A number

$$t_x=\cot^{-1} x,$$ (13)

where $x$ is an Integer or Rational Number, is sometimes called a Gregory Number. Lehmer (1938a) showed that $\cot^{-1} (a/b)$ can be expressed as a finite sum of inverse cotangents of Integer arguments

$$\cot^{-1}\left({a\over b}\right)= \sum_{i=1}^k (-1)^{i-1} \cot^{-1} n_i,$$ (14)

where

$$n_i=\left\lfloor{a_i\over b_i}\right\rfloor ,$$ (15)

with $\left\lfloor{x}\right\rfloor$ the Floor Function, and

$$a_{i+1} = a_in+i+b_i$$ (16)

$$b_{i+1} = a_i-n_ib_i,$$ (17)

with $a_0=a$ and $b_0=b$, and where the recurrence is continued until $b_{k+1}=0$. If an Inverse Tangent sum is written as

$$\tan^{-1} n=\sum_{k=1} f_k\tan^{-1} n_k+f\tan^{-1} 1,$$ (18)

then equation (14) becomes

$$\cot^{-1} n=\sum_{k=1} f_k\cot^{-1} n_k + c\cot^{-1} 1,$$ (19)

where

$$c=2-f-2\sum_{k=1} f_k.$$ (20)

Inverse cotangent sums can be used to generate Machin-Like Formulas.

An interesting inverse cotangent identity attributed to Charles Dodgson (Lewis Carroll) by Lehmer (1938b; Bromwich 1965, Castellanos 1988ab) is

$$\cot^{-1}(p+r)+\tan^{-1}(p+q)=\tan^{-1} p,$$ (21)

where

$$1+p^2=qr.$$ (22)

Other inverse cotangent identities include

$$2\cot^{-1}(2x)-\cot^{-1}x=\cot^{-1}(4x^3+3x)$$ (23)

$$3\cot^{-1}(3x)-\cot^{-1}x=\cot^{-1}\left({27x^4+18x^2-1\over 8x}\right),$$ (24)

as well as many others (Bennett 1926, Lehmer 1938b).

See also Cotangent, Inverse Tangent, Machin's Formula, Machin-Like Formulas, Tangent

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Inverse Circular Functions.'' §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.

Bennett, A. A. ``The Four Term Diophantine Arccotangent Relation.'' Ann. Math. 27, 21-24, 1926.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.

Bromwich, T. J. I. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988a.

Castellanos, D. ``The Ubiquitous Pi. Part II.'' Math. Mag. 61, 148-163, 1988b.

Lehmer, D. H. ``A Cotangent Analogue of Continued Fractions.'' Duke Math. J. 4, 323-340, 1938a.

Lehmer, D. H. ``On Arccotangent Relations for $\pi$.'' Amer. Math. Monthly 45, 657-664, 1938b.

Weisstein, E. W. ``Arccotangent Series.'' Mathematica notebook CotSeries.m.

Wetherfield, M. ``The Enhancement of Machin's Formula by Todd's Process.'' Math. Gaz., 333-344, July 1996.