§4.21 Identities

Keywords:: trigonometric functions
Permalink:: http://dlmf.nist.gov/4.21

§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

Notes:: See Hobson (1928, Chapter 4).
Keywords:: trigonometric functions
Permalink:: http://dlmf.nist.gov/4.21.i

4.21.1 $\mathop{\sin\/}\nolimits u\pm\mathop{\cos\/}\nolimits u=\sqrt{2}\mathop{\sin\/}\nolimits\!\left(u\pm\tfrac{1}{4}\pi\right)=\sqrt{2}\mathop{\cos\/}\nolimits\!\left(u\mp\tfrac{1}{4}\pi\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
Permalink:: http://dlmf.nist.gov/4.21.E1
Encodings:: TeX, pMML, png

4.21.2 $\mathop{\sin\/}\nolimits\!\left(u\pm v\right)=\mathop{\sin\/}\nolimits u\mathop{\cos\/}\nolimits v\pm\mathop{\cos\/}\nolimits u\mathop{\sin\/}\nolimits v,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.16
Permalink:: http://dlmf.nist.gov/4.21.E2
Encodings:: TeX, pMML, png

4.21.3 $\mathop{\cos\/}\nolimits\!\left(u\pm v\right)=\mathop{\cos\/}\nolimits u\mathop{\cos\/}\nolimits v\mp\mathop{\sin\/}\nolimits u\mathop{\sin\/}\nolimits v,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.17
Permalink:: http://dlmf.nist.gov/4.21.E3
Encodings:: TeX, pMML, png

4.21.4 $\mathop{\tan\/}\nolimits\!\left(u\pm v\right)=\frac{\mathop{\tan\/}\nolimits u\pm\mathop{\tan\/}\nolimits v}{1\mp\mathop{\tan\/}\nolimits u\mathop{\tan\/}\nolimits v},$

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function
A&S Ref:: 4.3.18
Permalink:: http://dlmf.nist.gov/4.21.E4
Encodings:: TeX, pMML, png

4.21.5 $\mathop{\cot\/}\nolimits\!\left(u\pm v\right)=\frac{\pm\mathop{\cot\/}\nolimits u\mathop{\cot\/}\nolimits v-1}{\mathop{\cot\/}\nolimits u\pm\mathop{\cot\/}\nolimits v}.$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function
A&S Ref:: 4.3.19
Permalink:: http://dlmf.nist.gov/4.21.E5
Encodings:: TeX, pMML, png

4.21.6 $\mathop{\sin\/}\nolimits u+\mathop{\sin\/}\nolimits v=2\mathop{\sin\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{\cos\/}\nolimits\!\left(\frac{u-v}{2}\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.34
Permalink:: http://dlmf.nist.gov/4.21.E6
Encodings:: TeX, pMML, png

4.21.7 $\mathop{\sin\/}\nolimits u-\mathop{\sin\/}\nolimits v=2\mathop{\cos\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{\sin\/}\nolimits\!\left(\frac{u-v}{2}\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.35
Permalink:: http://dlmf.nist.gov/4.21.E7
Encodings:: TeX, pMML, png

4.21.8 $\mathop{\cos\/}\nolimits u+\mathop{\cos\/}\nolimits v=2\mathop{\cos\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{\cos\/}\nolimits\!\left(\frac{u-v}{2}\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function
A&S Ref:: 4.3.36
Permalink:: http://dlmf.nist.gov/4.21.E8
Encodings:: TeX, pMML, png

4.21.9 $\mathop{\cos\/}\nolimits u-\mathop{\cos\/}\nolimits v=-2\mathop{\sin\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{\sin\/}\nolimits\!\left(\frac{u-v}{2}\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.37
Permalink:: http://dlmf.nist.gov/4.21.E9
Encodings:: TeX, pMML, png

4.21.10 $\mathop{\tan\/}\nolimits u\pm\mathop{\tan\/}\nolimits v=\frac{\mathop{\sin\/}\nolimits\!\left(u\pm v\right)}{\mathop{\cos\/}\nolimits u\mathop{\cos\/}\nolimits v},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $\mathop{\tan\/}\nolimits z$ : tangent function
A&S Ref:: 4.3.38
Permalink:: http://dlmf.nist.gov/4.21.E10
Encodings:: TeX, pMML, png

4.21.11 $\mathop{\cot\/}\nolimits u\pm\mathop{\cot\/}\nolimits v=\frac{\mathop{\sin\/}\nolimits\!\left(v\pm u\right)}{\mathop{\sin\/}\nolimits u\mathop{\sin\/}\nolimits v}.$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.39
Permalink:: http://dlmf.nist.gov/4.21.E11
Encodings:: TeX, pMML, png

§4.21(ii) Squares and Products

Notes:: See Hobson (1928, pp. 19, 45, 60).
Keywords:: trigonometric functions
Permalink:: http://dlmf.nist.gov/4.21.ii

4.21.12 ${\mathop{\sin\/}\nolimits^{{2}}}z+{\mathop{\cos\/}\nolimits^{{2}}}z=1,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.10
Permalink:: http://dlmf.nist.gov/4.21.E12
Encodings:: TeX, pMML, png

4.21.13 ${\mathop{\sec\/}\nolimits^{{2}}}z=1+{\mathop{\tan\/}\nolimits^{{2}}}z,$

Symbols:: $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\tan\/}\nolimits z$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.11
Permalink:: http://dlmf.nist.gov/4.21.E13
Encodings:: TeX, pMML, png

4.21.14 ${\mathop{\csc\/}\nolimits^{{2}}}z=1+{\mathop{\cot\/}\nolimits^{{2}}}z.$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cot\/}\nolimits z$ : cotangent function and $z$ : complex variable
A&S Ref:: 4.3.12
Permalink:: http://dlmf.nist.gov/4.21.E14
Encodings:: TeX, pMML, png

4.21.15 $2\mathop{\sin\/}\nolimits u\mathop{\sin\/}\nolimits v=\mathop{\cos\/}\nolimits\!\left(u-v\right)-\mathop{\cos\/}\nolimits\!\left(u+v\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.31
Permalink:: http://dlmf.nist.gov/4.21.E15
Encodings:: TeX, pMML, png

4.21.16 $2\mathop{\cos\/}\nolimits u\mathop{\cos\/}\nolimits v=\mathop{\cos\/}\nolimits\!\left(u-v\right)+\mathop{\cos\/}\nolimits\!\left(u+v\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function
A&S Ref:: 4.3.32
Permalink:: http://dlmf.nist.gov/4.21.E16
Encodings:: TeX, pMML, png

4.21.17 $2\mathop{\sin\/}\nolimits u\mathop{\cos\/}\nolimits v=\mathop{\sin\/}\nolimits\!\left(u-v\right)+\mathop{\sin\/}\nolimits\!\left(u+v\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.33
Permalink:: http://dlmf.nist.gov/4.21.E17
Encodings:: TeX, pMML, png

4.21.18 ${\mathop{\sin\/}\nolimits^{{2}}}u-{\mathop{\sin\/}\nolimits^{{2}}}v=\mathop{\sin\/}\nolimits\!\left(u+v\right)\mathop{\sin\/}\nolimits\!\left(u-v\right),$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.40
Permalink:: http://dlmf.nist.gov/4.21.E18
Encodings:: TeX, pMML, png

4.21.19 ${\mathop{\cos\/}\nolimits^{{2}}}u-{\mathop{\cos\/}\nolimits^{{2}}}v=-\mathop{\sin\/}\nolimits\!\left(u+v\right)\mathop{\sin\/}\nolimits\!\left(u-v\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.41
Permalink:: http://dlmf.nist.gov/4.21.E19
Encodings:: TeX, pMML, png

4.21.20 ${\mathop{\cos\/}\nolimits^{{2}}}u-{\mathop{\sin\/}\nolimits^{{2}}}v=\mathop{\cos\/}\nolimits\!\left(u+v\right)\mathop{\cos\/}\nolimits\!\left(u-v\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 4.3.42
Permalink:: http://dlmf.nist.gov/4.21.E20
Encodings:: TeX, pMML, png

§4.21(iii) Multiples of the Argument

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

4.21.21 $\mathop{\sin\/}\nolimits\frac{z}{2}=\pm\left(\frac{1-\mathop{\cos\/}\nolimits z}{2}\right)^{{1/2}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

4.21.22 $\mathop{\cos\/}\nolimits\frac{z}{2}=\pm\left(\frac{1+\mathop{\cos\/}\nolimits z}{2}\right)^{{1/2}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $z$ : complex variable
A&S Ref:: 4.3.21
Permalink:: http://dlmf.nist.gov/4.21.E22
Encodings:: TeX, pMML, png

4.21.23 $\mathop{\tan\/}\nolimits\frac{z}{2}=\pm\left(\frac{1-\mathop{\cos\/}\nolimits z}{1+\mathop{\cos\/}\nolimits z}\right)^{{1/2}}=\frac{1-\mathop{\cos\/}\nolimits z}{\mathop{\sin\/}\nolimits z}=\frac{\mathop{\sin\/}\nolimits z}{1+\mathop{\cos\/}\nolimits z}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : 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

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

4.21.24 $\mathop{\sin\/}\nolimits\!\left(-z\right)=-\mathop{\sin\/}\nolimits z,$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.13
Permalink:: http://dlmf.nist.gov/4.21.E24
Encodings:: TeX, pMML, png

4.21.25 $\mathop{\cos\/}\nolimits\!\left(-z\right)=\mathop{\cos\/}\nolimits z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $z$ : complex variable
A&S Ref:: 4.3.14
Permalink:: http://dlmf.nist.gov/4.21.E25
Encodings:: TeX, pMML, png

4.21.26 $\mathop{\tan\/}\nolimits\!\left(-z\right)=-\mathop{\tan\/}\nolimits z.$

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.15
Permalink:: http://dlmf.nist.gov/4.21.E26
Encodings:: TeX, pMML, png

4.21.27 $\mathop{\sin\/}\nolimits\!\left(2z\right)=2\mathop{\sin\/}\nolimits z\mathop{\cos\/}\nolimits z=\frac{2\mathop{\tan\/}\nolimits z}{1+{\mathop{\tan\/}\nolimits^{{2}}}z},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.24
Permalink:: http://dlmf.nist.gov/4.21.E27
Encodings:: TeX, pMML, png

4.21.28 $\mathop{\cos\/}\nolimits\!\left(2z\right)=2{\mathop{\cos\/}\nolimits^{{2}}}z-1=1-2{\mathop{\sin\/}\nolimits^{{2}}}z={\mathop{\cos\/}\nolimits^{{2}}}z-{\mathop{\sin\/}\nolimits^{{2}}}z=\frac{1-{\mathop{\tan\/}\nolimits^{{2}}}z}{1+{\mathop{\tan\/}\nolimits^{{2}}}z},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.25
Permalink:: http://dlmf.nist.gov/4.21.E28
Encodings:: TeX, pMML, png

4.21.29 $\mathop{\tan\/}\nolimits\!\left(2z\right)=\frac{2\mathop{\tan\/}\nolimits z}{1-{\mathop{\tan\/}\nolimits^{{2}}}z}=\frac{2\mathop{\cot\/}\nolimits z}{{\mathop{\cot\/}\nolimits^{{2}}}z-1}=\frac{2}{\mathop{\cot\/}\nolimits z-\mathop{\tan\/}\nolimits z}.$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\tan\/}\nolimits z$ : tangent function and $z$ : complex variable
A&S Ref:: 4.3.26
Permalink:: http://dlmf.nist.gov/4.21.E29
Encodings:: TeX, pMML, png

4.21.30 $\mathop{\sin\/}\nolimits\!\left(3z\right)=3\mathop{\sin\/}\nolimits z-4{\mathop{\sin\/}\nolimits^{{3}}}z,$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.27
Permalink:: http://dlmf.nist.gov/4.21.E30
Encodings:: TeX, pMML, png

4.21.31 $\mathop{\cos\/}\nolimits\!\left(3z\right)=-3\mathop{\cos\/}\nolimits z+4{\mathop{\cos\/}\nolimits^{{3}}}z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $z$ : complex variable
A&S Ref:: 4.3.28
Permalink:: http://dlmf.nist.gov/4.21.E31
Encodings:: TeX, pMML, png

4.21.32 $\mathop{\sin\/}\nolimits\!\left(4z\right)=8{\mathop{\cos\/}\nolimits^{{3}}}z\mathop{\sin\/}\nolimits z-4\mathop{\cos\/}\nolimits z\mathop{\sin\/}\nolimits z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 4.3.29
Permalink:: http://dlmf.nist.gov/4.21.E32
Encodings:: TeX, pMML, png

4.21.33 $\mathop{\cos\/}\nolimits\!\left(4z\right)=8{\mathop{\cos\/}\nolimits^{{4}}}z-8{\mathop{\cos\/}\nolimits^{{2}}}z+1.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $z$ : complex variable
A&S Ref:: 4.3.30
Permalink:: http://dlmf.nist.gov/4.21.E33
Encodings:: TeX, pMML, png

¶ De Moivre’s Theorem

Keywords:: De Moivre’s theorem, trigonometric functions

When $n\in\Integer$

4.21.34 $\mathop{\cos\/}\nolimits\!\left(nz\right)+i\mathop{\sin\/}\nolimits\!\left(nz\right)=(\mathop{\cos\/}\nolimits z+i\mathop{\sin\/}\nolimits z)^{n}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

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

4.21.35 $\mathop{\sin\/}\nolimits\!\left(nz\right)=2^{{n-1}}\prod _{{k=0}}^{{n-1}}\mathop{\sin\/}\nolimits\!\left(z+\frac{k\pi}{n}\right),$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{\sin\/}\nolimits z$ : 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

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

4.21.36

$\mathop{\sin\/}\nolimits z=\frac{2t}{1+t^{2}},$

$\mathop{\cos\/}\nolimits z=\frac{1-t^{2}}{1+t^{2}},$

$dz=\frac{2}{1+t^{2}}dt.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\sin\/}\nolimits z$ : 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

§4.21(iv) Real and Imaginary Parts; Moduli

Notes:: See Hobson (1928, p. 331).
Keywords:: trigonometric functions
Permalink:: http://dlmf.nist.gov/4.21.iv

With $z=x+iy$

4.21.37 $\mathop{\sin\/}\nolimits z=\mathop{\sin\/}\nolimits x\mathop{\cosh\/}\nolimits y+i\mathop{\cos\/}\nolimits x\mathop{\sinh\/}\nolimits y,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : 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

4.21.38 $\mathop{\cos\/}\nolimits z=\mathop{\cos\/}\nolimits x\mathop{\cosh\/}\nolimits y-i\mathop{\sin\/}\nolimits x\mathop{\sinh\/}\nolimits y,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : 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

4.21.39 $\mathop{\tan\/}\nolimits z=\frac{\mathop{\sin\/}\nolimits\!\left(2x\right)+i\mathop{\sinh\/}\nolimits\!\left(2y\right)}{\mathop{\cos\/}\nolimits\!\left(2x\right)+\mathop{\cosh\/}\nolimits\!\left(2y\right)},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : 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

4.21.40 $\mathop{\cot\/}\nolimits z=\frac{\mathop{\sin\/}\nolimits\!\left(2x\right)-i\mathop{\sinh\/}\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!\left(2y\right)-\mathop{\cos\/}\nolimits\!\left(2x\right)}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : 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

4.21.41 $|\mathop{\sin\/}\nolimits z|=({\mathop{\sin\/}\nolimits^{{2}}}x+{\mathop{\sinh\/}\nolimits^{{2}}}y)^{{1/2}}=\left(\tfrac{1}{2}\left(\mathop{\cosh\/}\nolimits\!\left(2y\right)-\mathop{\cos\/}\nolimits\!\left(2x\right)\right)\right)^{{1/2}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : 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

4.21.42 $|\mathop{\cos\/}\nolimits z|=({\mathop{\cos\/}\nolimits^{{2}}}x+{\mathop{\sinh\/}\nolimits^{{2}}}y)^{{1/2}}=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits\!\left(2y\right)+\mathop{\cos\/}\nolimits\!\left(2x\right))\right)^{{1/2}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : 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

4.21.43 $|\mathop{\tan\/}\nolimits z|=\left(\frac{\mathop{\cosh\/}\nolimits\!\left(2y\right)-\mathop{\cos\/}\nolimits\!\left(2x\right)}{\mathop{\cosh\/}\nolimits\!\left(2y\right)+\mathop{\cos\/}\nolimits\!\left(2x\right)}\right)^{{1/2}}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tan\/}\nolimits z$ : 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