# §4.21 Identities

## §4.21(i) Addition Formulas

 4.21.1 $\mathop{\sin\/}\nolimits u\pm\mathop{\cos\/}\nolimits u=\sqrt{2}\mathop{\sin\/% }\nolimits\!\left(u\pm\tfrac{1}{4}\pi\right)=\pm\sqrt{2}\mathop{\cos\/}% \nolimits\!\left(u\mp\tfrac{1}{4}\pi\right).$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function Referenced by: §4.21(i), Equation (4.21.1) Permalink: http://dlmf.nist.gov/4.21.E1 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the symbol $\pm$ was missing after the second equal sign. Reported 2012-09-27 by Dennis M. Heim See also: info for 4.21(i)
 4.21.2 $\displaystyle\mathop{\sin\/}\nolimits\!\left(u\pm v\right)$ $\displaystyle=\mathop{\sin\/}\nolimits u\mathop{\cos\/}\nolimits v\pm\mathop{% \cos\/}\nolimits u\mathop{\sin\/}\nolimits v,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.16 Referenced by: §4.21(i) Permalink: http://dlmf.nist.gov/4.21.E2 Encodings: TeX, pMML, png See also: info for 4.21(i) 4.21.3 $\displaystyle\mathop{\cos\/}\nolimits\!\left(u\pm v\right)$ $\displaystyle=\mathop{\cos\/}\nolimits u\mathop{\cos\/}\nolimits v\mp\mathop{% \sin\/}\nolimits u\mathop{\sin\/}\nolimits v,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(i) 4.21.4 $\displaystyle\mathop{\tan\/}\nolimits\!\left(u\pm v\right)$ $\displaystyle=\frac{\mathop{\tan\/}\nolimits u\pm\mathop{\tan\/}\nolimits v}{1% \mp\mathop{\tan\/}\nolimits u\mathop{\tan\/}\nolimits v},$ Symbols: $\mathop{\tan\/}\nolimits\NVar{z}$: tangent function A&S Ref: 4.3.18 Permalink: http://dlmf.nist.gov/4.21.E4 Encodings: TeX, pMML, png See also: info for 4.21(i) 4.21.5 $\displaystyle\mathop{\cot\/}\nolimits\!\left(u\pm v\right)$ $\displaystyle=\frac{\pm\mathop{\cot\/}\nolimits u\mathop{\cot\/}\nolimits v-1}% {\mathop{\cot\/}\nolimits u\pm\mathop{\cot\/}\nolimits v}.$ Symbols: $\mathop{\cot\/}\nolimits\NVar{z}$: cotangent function A&S Ref: 4.3.19 Permalink: http://dlmf.nist.gov/4.21.E5 Encodings: TeX, pMML, png See also: info for 4.21(i)
 4.21.6 $\displaystyle\mathop{\sin\/}\nolimits u+\mathop{\sin\/}\nolimits v$ $\displaystyle=2\mathop{\sin\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{% \cos\/}\nolimits\!\left(\frac{u-v}{2}\right),$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.34 Permalink: http://dlmf.nist.gov/4.21.E6 Encodings: TeX, pMML, png See also: info for 4.21(i) 4.21.7 $\displaystyle\mathop{\sin\/}\nolimits u-\mathop{\sin\/}\nolimits v$ $\displaystyle=2\mathop{\cos\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{% \sin\/}\nolimits\!\left(\frac{u-v}{2}\right),$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.35 Permalink: http://dlmf.nist.gov/4.21.E7 Encodings: TeX, pMML, png See also: info for 4.21(i) 4.21.8 $\displaystyle\mathop{\cos\/}\nolimits u+\mathop{\cos\/}\nolimits v$ $\displaystyle=2\mathop{\cos\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{% \cos\/}\nolimits\!\left(\frac{u-v}{2}\right),$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function A&S Ref: 4.3.36 Permalink: http://dlmf.nist.gov/4.21.E8 Encodings: TeX, pMML, png See also: info for 4.21(i) 4.21.9 $\displaystyle\mathop{\cos\/}\nolimits u-\mathop{\cos\/}\nolimits v$ $\displaystyle=-2\mathop{\sin\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{% \sin\/}\nolimits\!\left(\frac{u-v}{2}\right).$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.37 Permalink: http://dlmf.nist.gov/4.21.E9 Encodings: TeX, pMML, png See also: info for 4.21(i)
 4.21.10 $\displaystyle\mathop{\tan\/}\nolimits u\pm\mathop{\tan\/}\nolimits v$ $\displaystyle=\frac{\mathop{\sin\/}\nolimits\!\left(u\pm v\right)}{\mathop{% \cos\/}\nolimits u\mathop{\cos\/}\nolimits v},$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function and $\mathop{\tan\/}\nolimits\NVar{z}$: tangent function A&S Ref: 4.3.38 Permalink: http://dlmf.nist.gov/4.21.E10 Encodings: TeX, pMML, png See also: info for 4.21(i) 4.21.11 $\displaystyle\mathop{\cot\/}\nolimits u\pm\mathop{\cot\/}\nolimits v$ $\displaystyle=\frac{\mathop{\sin\/}\nolimits\!\left(v\pm u\right)}{\mathop{% \sin\/}\nolimits u\mathop{\sin\/}\nolimits v}.$ Symbols: $\mathop{\cot\/}\nolimits\NVar{z}$: cotangent function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.39 Permalink: http://dlmf.nist.gov/4.21.E11 Encodings: TeX, pMML, png See also: info for 4.21(i)

## §4.21(ii) Squares and Products

 4.21.12 ${\mathop{\sin\/}\nolimits^{2}}z+{\mathop{\cos\/}\nolimits^{2}}z=1,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(ii)
 4.21.13 ${\mathop{\sec\/}\nolimits^{2}}z=1+{\mathop{\tan\/}\nolimits^{2}}z,$ Symbols: $\mathop{\sec\/}\nolimits\NVar{z}$: secant function, $\mathop{\tan\/}\nolimits\NVar{z}$: 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: info for 4.21(ii)
 4.21.14 ${\mathop{\csc\/}\nolimits^{2}}z=1+{\mathop{\cot\/}\nolimits^{2}}z.$ Symbols: $\mathop{\csc\/}\nolimits\NVar{z}$: cosecant function, $\mathop{\cot\/}\nolimits\NVar{z}$: 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: info for 4.21(ii)
 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: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.31 Permalink: http://dlmf.nist.gov/4.21.E15 Encodings: TeX, pMML, png See also: info for 4.21(ii)
 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: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function A&S Ref: 4.3.32 Permalink: http://dlmf.nist.gov/4.21.E16 Encodings: TeX, pMML, png See also: info for 4.21(ii)
 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: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.33 Permalink: http://dlmf.nist.gov/4.21.E17 Encodings: TeX, pMML, png See also: info for 4.21(ii)
 4.21.18 $\displaystyle{\mathop{\sin\/}\nolimits^{2}}u-{\mathop{\sin\/}\nolimits^{2}}v$ $\displaystyle=\mathop{\sin\/}\nolimits\!\left(u+v\right)\mathop{\sin\/}% \nolimits\!\left(u-v\right),$ Symbols: $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.40 Permalink: http://dlmf.nist.gov/4.21.E18 Encodings: TeX, pMML, png See also: info for 4.21(ii) 4.21.19 $\displaystyle{\mathop{\cos\/}\nolimits^{2}}u-{\mathop{\cos\/}\nolimits^{2}}v$ $\displaystyle=-\mathop{\sin\/}\nolimits\!\left(u+v\right)\mathop{\sin\/}% \nolimits\!\left(u-v\right),$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.41 Permalink: http://dlmf.nist.gov/4.21.E19 Encodings: TeX, pMML, png See also: info for 4.21(ii) 4.21.20 $\displaystyle{\mathop{\cos\/}\nolimits^{2}}u-{\mathop{\sin\/}\nolimits^{2}}v$ $\displaystyle=\mathop{\cos\/}\nolimits\!\left(u+v\right)\mathop{\cos\/}% \nolimits\!\left(u-v\right).$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function A&S Ref: 4.3.42 Permalink: http://dlmf.nist.gov/4.21.E20 Encodings: TeX, pMML, png See also: info for 4.21(ii)

## §4.21(iii) Multiples of the Argument

 4.21.21 $\mathop{\sin\/}\nolimits\frac{z}{2}=\pm\left(\frac{1-\mathop{\cos\/}\nolimits z% }{2}\right)^{1/2},$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)
 4.21.22 $\mathop{\cos\/}\nolimits\frac{z}{2}=\pm\left(\frac{1+\mathop{\cos\/}\nolimits z% }{2}\right)^{1/2},$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)
 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: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $\mathop{\tan\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)

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

 4.21.24 $\displaystyle\mathop{\sin\/}\nolimits\!\left(-z\right)$ $\displaystyle=-\mathop{\sin\/}\nolimits z,$ Symbols: $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(iii) 4.21.25 $\displaystyle\mathop{\cos\/}\nolimits\!\left(-z\right)$ $\displaystyle=\mathop{\cos\/}\nolimits z,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: 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: info for 4.21(iii) 4.21.26 $\displaystyle\mathop{\tan\/}\nolimits\!\left(-z\right)$ $\displaystyle=-\mathop{\tan\/}\nolimits z.$ Symbols: $\mathop{\tan\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)
 4.21.27 $\mathop{\sin\/}\nolimits\!\left(2z\right)=2\mathop{\sin\/}\nolimits z\mathop{% \cos\/}\nolimits z=\frac{2\mathop{\tan\/}\nolimits z}{1+{\mathop{\tan\/}% \nolimits^{2}}z},$
 4.21.28 $\mathop{\cos\/}\nolimits\!\left(2z\right)=2{\mathop{\cos\/}\nolimits^{2}}z-1=1% -2{\mathop{\sin\/}\nolimits^{2}}z={\mathop{\cos\/}\nolimits^{2}}z-{\mathop{% \sin\/}\nolimits^{2}}z=\frac{1-{\mathop{\tan\/}\nolimits^{2}}z}{1+{\mathop{% \tan\/}\nolimits^{2}}z},$
 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: $\mathop{\cot\/}\nolimits\NVar{z}$: cotangent function, $\mathop{\tan\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)
 4.21.30 $\displaystyle\mathop{\sin\/}\nolimits\!\left(3z\right)$ $\displaystyle=3\mathop{\sin\/}\nolimits z-4{\mathop{\sin\/}\nolimits^{3}}z,$ Symbols: $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(iii) 4.21.31 $\displaystyle\mathop{\cos\/}\nolimits\!\left(3z\right)$ $\displaystyle=-3\mathop{\cos\/}\nolimits z+4{\mathop{\cos\/}\nolimits^{3}}z,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: 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: info for 4.21(iii) 4.21.32 $\displaystyle\mathop{\sin\/}\nolimits\!\left(4z\right)$ $\displaystyle=8{\mathop{\cos\/}\nolimits^{3}}z\mathop{\sin\/}\nolimits z-4% \mathop{\cos\/}\nolimits z\mathop{\sin\/}\nolimits z,$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(iii) 4.21.33 $\displaystyle\mathop{\cos\/}\nolimits\!\left(4z\right)$ $\displaystyle=8{\mathop{\cos\/}\nolimits^{4}}z-8{\mathop{\cos\/}\nolimits^{2}}% z+1.$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)

### De Moivre’s Theorem

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: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)

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: $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)

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

 4.21.36 $\displaystyle\mathop{\sin\/}\nolimits z$ $\displaystyle=\frac{2t}{1+t^{2}},$ $\displaystyle\mathop{\cos\/}\nolimits z$ $\displaystyle=\frac{1-t^{2}}{1+t^{2}},$ $\displaystyle dz$ $\displaystyle=\frac{2}{1+t^{2}}dt.$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $d\NVar{x}$: differential of $x$, $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.21(iii)

## §4.21(iv) Real and Imaginary Parts; Moduli

With $z=x+iy$

 4.21.37 $\mathop{\sin\/}\nolimits z=\mathop{\sin\/}\nolimits x\mathop{\cosh\/}\nolimits y% +i\mathop{\cos\/}\nolimits x\mathop{\sinh\/}\nolimits y,$
 4.21.38 $\mathop{\cos\/}\nolimits z=\mathop{\cos\/}\nolimits x\mathop{\cosh\/}\nolimits y% -i\mathop{\sin\/}\nolimits x\mathop{\sinh\/}\nolimits y,$
 4.21.39 $\mathop{\tan\/}\nolimits z=\frac{\mathop{\sin\/}\nolimits\!\left(2x\right)+i% \mathop{\sinh\/}\nolimits\!\left(2y\right)}{\mathop{\cos\/}\nolimits\!\left(2x% \right)+\mathop{\cosh\/}\nolimits\!\left(2y\right)},$
 4.21.40 $\mathop{\cot\/}\nolimits z=\frac{\mathop{\sin\/}\nolimits\!\left(2x\right)-i% \mathop{\sinh\/}\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!\left(2% y\right)-\mathop{\cos\/}\nolimits\!\left(2x\right)}.$
 4.21.41 $|\mathop{\sin\/}\nolimits z|=({\mathop{\sin\/}\nolimits^{2}}x+{\mathop{\sinh\/% }\nolimits^{2}}y)^{1/2}=\left(\tfrac{1}{2}\left(\mathop{\cosh\/}\nolimits\!% \left(2y\right)-\mathop{\cos\/}\nolimits\!\left(2x\right)\right)\right)^{1/2},$
 4.21.42 $|\mathop{\cos\/}\nolimits z|=({\mathop{\cos\/}\nolimits^{2}}x+{\mathop{\sinh\/% }\nolimits^{2}}y)^{1/2}=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits\!\left(2y% \right)+\mathop{\cos\/}\nolimits\!\left(2x\right))\right)^{1/2},$
 4.21.43 $|\mathop{\tan\/}\nolimits z|=\left(\frac{\mathop{\cosh\/}\nolimits\!\left(2y% \right)-\mathop{\cos\/}\nolimits\!\left(2x\right)}{\mathop{\cosh\/}\nolimits\!% \left(2y\right)+\mathop{\cos\/}\nolimits\!\left(2x\right)}\right)^{1/2}.$