# §4.21 Identities

###### Contents

 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), 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: Annotations for 4.21(i), 4.21 and 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 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 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 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 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 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 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 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 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 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 4

## §4.21(ii) Squares and Products

 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 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 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 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 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 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 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 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 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 4

## §4.21(iii) Multiples of the Argument

 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 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 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 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 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 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 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 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 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 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 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 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 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 4

### De Moivre’s Theorem

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, $\sin\NVar{z}$: 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 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 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 4

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

With $z=x+iy$

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