4 Elementary Functions Trigonometric Functions 4.20 Derivatives and Differential Equations 4.22 Infinite Products and Partial Fractions

§4.21 Identities

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Keywords:: identities, trigonometric functions
Permalink:: http://dlmf.nist.gov/4.21
See also:: Annotations for Ch.4

§4.21(i) Addition Formulas
§4.21(ii) Squares and Products
§4.21(iii) Multiples of the Argument
§4.21(iv) Real and Imaginary Parts; Moduli

§4.21(i) Addition Formulas

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Keywords:: addition formulas, trigonometric functions
Notes:: See Hobson (1928, Chapter 4). For (4.21.1) use (4.21.2) and (4.21.3) with $\nu=\tfrac{1}{4}\pi$ .
Permalink:: http://dlmf.nist.gov/4.21.i
Addition (effective with 1.1.0):: Equation (4.21.1_5) was added.
Suggested 2018-08-14 by Ted Ersek
See also:: Annotations for §4.21 and Ch.4

4.21.1		$\sin u\pm\cos u=\sqrt{2}\sin\left(u\pm\tfrac{1}{4}\pi\right)=\pm\sqrt{2}\cos% \left(u\mp\tfrac{1}{4}\pi\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function Referenced by: §4.21(i), Erratum (V1.0.7) for Equation (4.21.1) Permalink: http://dlmf.nist.gov/4.21.E1 Encodings: TeX, pMML, png Correction (effective with 1.0.7): Originally the symbol $\pm$ was missing after the second equal sign. Suggested 2012-09-27 by Dennis M. Heim See also: Annotations for §4.21(i), §4.21 and Ch.4

4.21.1_5		$A\cos u+B\sin u=\sqrt{A^{2}+B^{2}}\cos\left(u-\operatorname{ph}\left(A+B% \mathrm{i}\right)\right),$
$A,B\in\mathbb{R}$ ,
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $\mathbb{R}$ : real line and $\sin\NVar{z}$ : sine function Referenced by: §4.21(i), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/4.21.E1_5 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. Suggested 2018-08-14 by Ted Ersek See also: Annotations for §4.21(i), §4.21 and Ch.4

4.21.2	$\displaystyle\sin\left(u\pm v\right)$	$\displaystyle=\sin u\cos v\pm\cos u\sin v,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.16 Referenced by: §4.21(i), (9.8.13) Permalink: http://dlmf.nist.gov/4.21.E2 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.3	$\displaystyle\cos\left(u\pm v\right)$	$\displaystyle=\cos u\cos v\mp\sin u\sin v,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.17 Referenced by: §4.21(i) Permalink: http://dlmf.nist.gov/4.21.E3 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.4	$\displaystyle\tan\left(u\pm v\right)$	$\displaystyle=\frac{\tan u\pm\tan v}{1\mp\tan u\tan v},$
ⓘ Symbols: $\tan\NVar{z}$ : tangent function A&S Ref: 4.3.18 Referenced by: (9.8.17) Permalink: http://dlmf.nist.gov/4.21.E4 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.5	$\displaystyle\cot\left(u\pm v\right)$	$\displaystyle=\frac{\pm\cot u\cot v-1}{\cot u\pm\cot v}.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function A&S Ref: 4.3.19 Permalink: http://dlmf.nist.gov/4.21.E5 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4

4.21.6	$\displaystyle\sin u+\sin v$	$\displaystyle=2\sin\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.34 Permalink: http://dlmf.nist.gov/4.21.E6 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.7	$\displaystyle\sin u-\sin v$	$\displaystyle=2\cos\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.35 Permalink: http://dlmf.nist.gov/4.21.E7 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.8	$\displaystyle\cos u+\cos v$	$\displaystyle=2\cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function A&S Ref: 4.3.36 Permalink: http://dlmf.nist.gov/4.21.E8 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.9	$\displaystyle\cos u-\cos v$	$\displaystyle=-2\sin\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.37 Permalink: http://dlmf.nist.gov/4.21.E9 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4

4.21.10	$\displaystyle\tan u\pm\tan v$	$\displaystyle=\frac{\sin\left(u\pm v\right)}{\cos u\cos v},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $\tan\NVar{z}$ : tangent function A&S Ref: 4.3.38 Permalink: http://dlmf.nist.gov/4.21.E10 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4
4.21.11	$\displaystyle\cot u\pm\cot v$	$\displaystyle=\frac{\sin\left(v\pm u\right)}{\sin u\sin v}.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.39 Permalink: http://dlmf.nist.gov/4.21.E11 Encodings: TeX, pMML, png See also: Annotations for §4.21(i), §4.21 and Ch.4

§4.21(ii) Squares and Products

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Keywords:: squares and products, trigonometric functions
Notes:: See Hobson (1928, pp. 19, 45, 60).
Permalink:: http://dlmf.nist.gov/4.21.ii
See also:: Annotations for §4.21 and Ch.4

4.21.12		${\sin}^{2}z+{\cos}^{2}z=1,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.10 Permalink: http://dlmf.nist.gov/4.21.E12 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.13		${\sec}^{2}z=1+{\tan}^{2}z,$
ⓘ Symbols: $\sec\NVar{z}$ : secant function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.11 Permalink: http://dlmf.nist.gov/4.21.E13 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.14		${\csc}^{2}z=1+{\cot}^{2}z.$
ⓘ Symbols: $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function and $z$ : complex variable A&S Ref: 4.3.12 Permalink: http://dlmf.nist.gov/4.21.E14 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.15		$2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.31 Permalink: http://dlmf.nist.gov/4.21.E15 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.16		$2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function A&S Ref: 4.3.32 Permalink: http://dlmf.nist.gov/4.21.E16 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.17		$2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v\right).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.33 Permalink: http://dlmf.nist.gov/4.21.E17 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

4.21.18	$\displaystyle{\sin}^{2}u-{\sin}^{2}v$	$\displaystyle=\sin\left(u+v\right)\sin\left(u-v\right),$
ⓘ Symbols: $\sin\NVar{z}$ : sine function A&S Ref: 4.3.40 Permalink: http://dlmf.nist.gov/4.21.E18 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4
4.21.19	$\displaystyle{\cos}^{2}u-{\cos}^{2}v$	$\displaystyle=-\sin\left(u+v\right)\sin\left(u-v\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.41 Permalink: http://dlmf.nist.gov/4.21.E19 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4
4.21.20	$\displaystyle{\cos}^{2}u-{\sin}^{2}v$	$\displaystyle=\cos\left(u+v\right)\cos\left(u-v\right).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.42 Permalink: http://dlmf.nist.gov/4.21.E20 Encodings: TeX, pMML, png See also: Annotations for §4.21(ii), §4.21 and Ch.4

§4.21(iii) Multiples of the Argument

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Keywords:: multiples of argument, trigonometric functions
Notes:: See Hobson (1928, pp. 21, 52–53, 63–69, 237–239). For (4.21.35) see Walker (1996, §1.9).
Permalink:: http://dlmf.nist.gov/4.21.iii
See also:: Annotations for §4.21 and Ch.4

4.21.21		$\sin\frac{z}{2}=\pm\left(\frac{1-\cos z}{2}\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.20 Referenced by: §4.21(iii) Permalink: http://dlmf.nist.gov/4.21.E21 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.22		$\cos\frac{z}{2}=\pm\left(\frac{1+\cos z}{2}\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 4.3.21 Permalink: http://dlmf.nist.gov/4.21.E22 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.23		$\tan\frac{z}{2}=\pm\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=\frac{1-\cos z% }{\sin z}=\frac{\sin z}{1+\cos z}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.22 Referenced by: §4.21(iii) Permalink: http://dlmf.nist.gov/4.21.E23 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

In (4.21.21)–(4.21.23) Table 4.16.1 and analytic continuation will assist in resolving sign ambiguities.

4.21.24	$\displaystyle\sin\left(-z\right)$	$\displaystyle=-\sin z,$
ⓘ Symbols: $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.13 Permalink: http://dlmf.nist.gov/4.21.E24 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.25	$\displaystyle\cos\left(-z\right)$	$\displaystyle=\cos z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 4.3.14 Permalink: http://dlmf.nist.gov/4.21.E25 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.26	$\displaystyle\tan\left(-z\right)$	$\displaystyle=-\tan z.$
ⓘ Symbols: $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.15 Permalink: http://dlmf.nist.gov/4.21.E26 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.27		$\sin\left(2z\right)=2\sin z\cos z=\frac{2\tan z}{1+{\tan}^{2}z},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.24 Permalink: http://dlmf.nist.gov/4.21.E27 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.28		$\cos\left(2z\right)=2{\cos}^{2}z-1=1-2{\sin}^{2}z={\cos}^{2}z-{\sin}^{2}z=% \frac{1-{\tan}^{2}z}{1+{\tan}^{2}z},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.25 Permalink: http://dlmf.nist.gov/4.21.E28 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.29		$\tan\left(2z\right)=\frac{2\tan z}{1-{\tan}^{2}z}=\frac{2\cot z}{{\cot}^{2}z-1% }=\frac{2}{\cot z-\tan z}.$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 4.3.26 Permalink: http://dlmf.nist.gov/4.21.E29 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

4.21.30	$\displaystyle\sin\left(3z\right)$	$\displaystyle=3\sin z-4{\sin}^{3}z,$
ⓘ Symbols: $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.27 Permalink: http://dlmf.nist.gov/4.21.E30 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.31	$\displaystyle\cos\left(3z\right)$	$\displaystyle=-3\cos z+4{\cos}^{3}z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 4.3.28 Permalink: http://dlmf.nist.gov/4.21.E31 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.32	$\displaystyle\sin\left(4z\right)$	$\displaystyle=8{\cos}^{3}z\sin z-4\cos z\sin z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.29 Permalink: http://dlmf.nist.gov/4.21.E32 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4
4.21.33	$\displaystyle\cos\left(4z\right)$	$\displaystyle=8{\cos}^{4}z-8{\cos}^{2}z+1.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 4.3.30 Permalink: http://dlmf.nist.gov/4.21.E33 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21 and Ch.4

De Moivre’s Theorem

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Keywords:: De Moivre’s theorem, trigonometric functions
See also:: Annotations for §4.21(iii), §4.21 and Ch.4

When $n\in\mathbb{Z}$

4.21.34		$\cos\left(nz\right)+i\sin\left(nz\right)=(\cos z+i\sin z)^{n}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable A&S Ref: 4.3.48 Permalink: http://dlmf.nist.gov/4.21.E34 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21(iii), §4.21 and Ch.4

This result is also valid when $n$ is fractional or complex, provided that $-\pi\leq\Re z\leq\pi$ .

4.21.35		$\sin\left(nz\right)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(z+\frac{k\pi}{n}\right),$
$n=1,2,3,\dots$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $k$ : integer, $n$ : integer and $z$ : complex variable Referenced by: §23.10(iii), §4.21(iii) Permalink: http://dlmf.nist.gov/4.21.E35 Encodings: TeX, pMML, png See also: Annotations for §4.21(iii), §4.21(iii), §4.21 and Ch.4

If $t=\tan\left(\frac{1}{2}z\right)$ , then

4.21.36	$\displaystyle\sin z$	$\displaystyle=\frac{2t}{1+t^{2}},$
	$\displaystyle\cos z$	$\displaystyle=\frac{1-t^{2}}{1+t^{2}},$
	$\displaystyle\,\mathrm{d}z$	$\displaystyle=\frac{2}{1+t^{2}}\,\mathrm{d}t.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 4.3.23 Permalink: http://dlmf.nist.gov/4.21.E36 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §4.21(iii), §4.21(iii), §4.21 and Ch.4

§4.21(iv) Real and Imaginary Parts; Moduli

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Keywords:: moduli, real and imaginary parts, trigonometric functions
Notes:: See Hobson (1928, p. 331).
Permalink:: http://dlmf.nist.gov/4.21.iv
See also:: Annotations for §4.21 and Ch.4

With $z=x+iy$

4.21.37		$\sin z=\sin x\cosh y+\mathrm{i}\cos x\sinh y,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.55 Permalink: http://dlmf.nist.gov/4.21.E37 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.38		$\cos z=\cos x\cosh y-\mathrm{i}\sin x\sinh y,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.56 Permalink: http://dlmf.nist.gov/4.21.E38 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.39		$\tan z=\frac{\sin\left(2x\right)+\mathrm{i}\sinh\left(2y\right)}{\cos\left(2x% \right)+\cosh\left(2y\right)},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.57 Permalink: http://dlmf.nist.gov/4.21.E39 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.40		$\cot z=\frac{\sin\left(2x\right)-\mathrm{i}\sinh\left(2y\right)}{\cosh\left(2y% \right)-\cos\left(2x\right)}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.58 Permalink: http://dlmf.nist.gov/4.21.E40 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.41		$\|\sin z\|=({\sin}^{2}x+{\sinh}^{2}y)^{1/2}=\left(\tfrac{1}{2}\left(\cosh\left(2% y\right)-\cos\left(2x\right)\right)\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.59 Permalink: http://dlmf.nist.gov/4.21.E41 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.42		$\|\cos z\|=({\cos}^{2}x+{\sinh}^{2}y)^{1/2}=\left(\tfrac{1}{2}(\cosh\left(2y% \right)+\cos\left(2x\right))\right)^{1/2},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.61 Permalink: http://dlmf.nist.gov/4.21.E42 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.21.43		$\|\tan z\|=\left(\frac{\cosh\left(2y\right)-\cos\left(2x\right)}{\cosh\left(2y% \right)+\cos\left(2x\right)}\right)^{1/2}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\tan\NVar{z}$ : tangent function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.3.63 Permalink: http://dlmf.nist.gov/4.21.E43 Encodings: TeX, pMML, png See also: Annotations for §4.21(iv), §4.21 and Ch.4

4.20 Derivatives and Differential Equations 4.22 Infinite Products and Partial Fractions

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