Alla trigonometriska funktioner för en vinkel θ kan konstrueras geometriskt i termer av en enhetscirkel med centrum i punkten O Lista över trigonometriska identiteter är en lista av ekvationer som involverar trigonometriska funktioner och som är sanna för varje enskilt värde av de förekommande variablerna. De skiljer sig från triangelidentiteter, vilka är identiteter som potentiellt involverar vinklar, men även omfattar sidolängder eller andra längder i en triangel. Endast de förstnämnda behandlas i denna artikel.
Identiteterna är användbara när uttryck som involverar trigonometriska funktioner måste förenklas. En viktig tillämpning är integration av icke-trigonometriska funktioner: en vanlig teknik är att först göra en substitution med en trigonometrisk funktion och sedan förenkla resultatet med hjälp av en trigonometrisk identitet.
Grundläggande
cos
(
x
)
=
sin
(
x
+
π
2
)
{\displaystyle \cos(x)=\sin \left(x+{\frac {\pi }{2}}\right)}
tan
(
x
)
=
sin
(
x
)
cos
(
x
)
{\displaystyle \tan(x)={\frac {\sin(x)}{\cos(x)}}}
cot
(
x
)
=
cos
(
x
)
sin
(
x
)
=
tan
(
π
2
−
x
)
{\displaystyle \cot(x)={\frac {\cos(x)}{\sin(x)}}=\tan({\frac {\pi }{2}}-x)}
sec
(
x
)
=
1
cos
(
x
)
{\displaystyle \sec(x)={\frac {1}{\cos(x)}}}
csc
(
x
)
=
1
sin
(
x
)
{\displaystyle \csc(x)={\frac {1}{\sin(x)}}}
Sinus, cosinus, sekant och cosekant har perioden 2π. Tangens och cotangens har perioden π. Om k är ett heltal gäller:
sin
(
x
)
=
sin
(
x
+
2
k
π
)
cos
(
x
)
=
cos
(
x
+
2
k
π
)
tan
(
x
)
=
tan
(
x
+
k
π
)
cot
(
x
)
=
cot
(
x
+
k
π
)
sec
(
x
)
=
sec
(
x
+
2
k
π
)
csc
(
x
)
=
csc
(
x
+
2
k
π
)
{\displaystyle {\begin{aligned}\sin(x)&=\sin(x+2k\pi )\\\cos(x)&=\cos(x+2k\pi )\\\tan(x)&=\tan(x+k\pi )\\\cot(x)&=\cot(x+k\pi )\\\sec(x)&=\sec(x+2k\pi )\\\csc(x)&=\csc(x+2k\pi )\\\end{aligned}}}
sin
(
−
x
)
=
−
sin
(
x
)
sin
(
π
2
−
x
)
=
cos
(
x
)
sin
(
π
−
x
)
=
+
sin
(
x
)
cos
(
−
x
)
=
+
cos
(
x
)
cos
(
π
2
−
x
)
=
sin
(
x
)
cos
(
π
−
x
)
=
−
cos
(
x
)
tan
(
−
x
)
=
−
tan
(
x
)
tan
(
π
2
−
x
)
=
cot
(
x
)
tan
(
π
−
x
)
=
−
tan
(
x
)
cot
(
−
x
)
=
−
cot
(
x
)
cot
(
π
2
−
x
)
=
tan
(
x
)
cot
(
π
−
x
)
=
−
cot
(
x
)
sec
(
−
x
)
=
+
sec
(
x
)
sec
(
π
2
−
x
)
=
csc
(
x
)
sec
(
π
−
x
)
=
−
sec
(
x
)
csc
(
−
x
)
=
−
csc
(
x
)
csc
(
π
2
−
x
)
=
sec
(
x
)
csc
(
π
−
x
)
=
+
csc
(
x
)
{\displaystyle {\begin{aligned}\sin(-x)&=-\sin(x)&\sin \left({\cfrac {\pi }{2}}-x\right)&=\cos(x)&\sin \left(\pi -x\right)&=+\sin(x)\\\cos(-x)&=+\cos(x)&\cos \left({\cfrac {\pi }{2}}-x\right)&=\sin(x)&\cos \left(\pi -x\right)&=-\cos(x)\\\tan(-x)&=-\tan(x)&\tan \left({\cfrac {\pi }{2}}-x\right)&=\cot(x)&\tan \left(\pi -x\right)&=-\tan(x)\\\cot(-x)&=-\cot(x)&\cot \left({\cfrac {\pi }{2}}-x\right)&=\tan(x)&\cot \left(\pi -x\right)&=-\cot(x)\\\sec(-x)&=+\sec(x)&\sec \left({\cfrac {\pi }{2}}-x\right)&=\csc(x)&\sec \left(\pi -x\right)&=-\sec(x)\\\csc(-x)&=-\csc(x)&\csc \left({\cfrac {\pi }{2}}-x\right)&=\sec(x)&\csc \left(\pi -x\right)&=+\csc(x)\\\end{aligned}}}
En funktion f(x) kallas udda om f(-x) = -f(x) och kallas jämn om f(-x) = f(x). Till exempel är cosinusfunktionen jämn och sinus- och tangensfunktionerna är udda.
Förskjutningar
sin
(
x
+
π
2
)
=
+
cos
(
x
)
sin
(
x
+
π
)
=
−
sin
(
x
)
cos
(
x
+
π
2
)
=
−
sin
(
x
)
cos
(
x
+
π
)
=
−
cos
(
x
)
tan
(
x
+
π
2
)
=
−
cot
(
x
)
tan
(
x
+
π
)
=
+
tan
(
x
)
{\displaystyle {\begin{aligned}\sin \left(x+{\cfrac {\pi }{2}}\right)&=+\cos(x)&\sin \left(x+\pi \right)&=-\sin(x)\\\cos \left(x+{\cfrac {\pi }{2}}\right)&=-\sin(x)&\cos \left(x+\pi \right)&=-\cos(x)\\\tan \left(x+{\cfrac {\pi }{2}}\right)&=-\cot(x)&\tan \left(x+\pi \right)&=+\tan(x)\\\end{aligned}}}
Samband för en vinkel
sin
2
(
x
)
+
cos
2
(
x
)
=
1
{\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}
sin
(
x
)
=
±
1
−
cos
2
(
x
)
{\displaystyle \sin(x)=\pm {\sqrt {1-\cos ^{2}(x)}}}
cos
(
x
)
=
±
1
−
sin
2
(
x
)
{\displaystyle \cos(x)=\pm {\sqrt {1-\sin ^{2}(x)}}}
Relaterade identiteter
1
+
tan
2
θ
=
sec
2
θ
{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }
1
+
cot
2
θ
=
csc
2
θ
{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }
Dubbla vinkeln
sin
(
2
x
)
=
2
sin
(
x
)
cos
(
x
)
cos
(
2
x
)
=
cos
2
(
x
)
−
sin
2
(
x
)
=
=
2
cos
2
(
x
)
−
1
=
=
1
−
2
sin
2
(
x
)
tan
(
2
x
)
=
2
tan
(
x
)
1
−
tan
2
(
x
)
cot
(
2
x
)
=
cot
(
x
)
−
tan
(
x
)
2
{\displaystyle {\begin{aligned}\sin(2x)&=2\sin(x)\cos(x)\\\cos(2x)&=\cos ^{2}(x)-\sin ^{2}(x)=\\&=2\cos ^{2}(x)-1=\\&=1-2\sin ^{2}(x)\\\tan(2x)&={\frac {2\tan(x)}{1-\tan ^{2}(x)}}\\\cot(2x)&={\frac {\cot(x)-\tan(x)}{2}}\\\end{aligned}}}
Tredubbla vinkeln
sin
(
3
x
)
=
3
sin
(
x
)
−
4
sin
3
(
x
)
cos
(
3
x
)
=
4
cos
3
(
x
)
−
3
cos
(
x
)
tan
(
3
x
)
=
3
tan
(
x
)
−
tan
3
(
x
)
1
−
3
tan
2
(
x
)
cot
(
3
x
)
=
cot
3
(
x
)
−
3
cot
(
x
)
3
cot
2
(
x
)
−
1
{\displaystyle {\begin{aligned}\sin(3x)&=3\sin(x)-4\sin ^{3}(x)\\\cos(3x)&=4\cos ^{3}(x)-3\cos(x)\\\tan(3x)&={\frac {3\tan(x)-\tan ^{3}(x)}{1-3\tan ^{2}(x)}}\\\cot(3x)&={\frac {\cot ^{3}(x)-3\cot(x)}{3\cot ^{2}(x)-1}}\\\end{aligned}}}
Halva vinkeln
sin
2
(
x
2
)
=
1
−
cos
(
x
)
2
cos
2
(
x
2
)
=
1
+
cos
(
x
)
2
tan
(
x
2
)
=
sin
(
x
)
1
+
cos
(
x
)
=
=
1
−
cos
(
x
)
sin
(
x
)
tan
2
(
x
2
)
=
1
−
cos
(
x
)
1
+
cos
(
x
)
cot
(
x
2
)
=
sin
(
x
)
1
−
cos
(
x
)
=
=
1
+
cos
(
x
)
sin
(
x
)
cot
2
(
x
2
)
=
1
+
cos
(
x
)
1
−
cos
(
x
)
{\displaystyle {\begin{aligned}\sin ^{2}\left({\frac {x}{2}}\right)&={\frac {1-\cos(x)}{2}}\\\cos ^{2}\left({\frac {x}{2}}\right)&={\frac {1+\cos(x)}{2}}\\\tan \left({\frac {x}{2}}\right)&={\frac {\sin(x)}{1+\cos(x)}}&=\\&={\frac {1-\cos(x)}{\sin(x)}}\\\tan ^{2}\left({\frac {x}{2}}\right)&={\frac {1-\cos(x)}{1+\cos(x)}}\\\cot \left({\frac {x}{2}}\right)&={\frac {\sin(x)}{1-\cos(x)}}&=\\&={\frac {1+\cos(x)}{\sin(x)}}\\\cot ^{2}\left({\frac {x}{2}}\right)&={\frac {1+\cos(x)}{1-\cos(x)}}\\\end{aligned}}}
Potenser
sin
2
θ
=
1
−
cos
2
θ
2
{\displaystyle \sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}}
cos
2
θ
=
1
+
cos
2
θ
2
{\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}}
sin
2
θ
cos
2
θ
=
1
−
cos
4
θ
8
{\displaystyle \sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos 4\theta }{8}}}
sin
3
θ
=
3
sin
θ
−
sin
3
θ
4
{\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin 3\theta }{4}}}
cos
3
θ
=
3
cos
θ
+
cos
3
θ
4
{\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos 3\theta }{4}}}
sin
3
θ
cos
3
θ
=
3
sin
2
θ
−
sin
6
θ
32
{\displaystyle \sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin 2\theta -\sin 6\theta }{32}}}
sin
4
θ
=
3
−
4
cos
2
θ
+
cos
4
θ
8
{\displaystyle \sin ^{4}\theta ={\frac {3-4\cos 2\theta +\cos 4\theta }{8}}}
cos
4
θ
=
3
+
4
cos
2
θ
+
cos
4
θ
8
{\displaystyle \cos ^{4}\theta ={\frac {3+4\cos 2\theta +\cos 4\theta }{8}}}
sin
4
θ
cos
4
θ
=
3
−
4
cos
4
θ
+
cos
8
θ
128
{\displaystyle \sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos 4\theta +\cos 8\theta }{128}}}
sin
5
θ
=
10
sin
θ
−
5
sin
3
θ
+
sin
5
θ
16
{\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin 3\theta +\sin 5\theta }{16}}}
cos
5
θ
=
10
cos
θ
+
5
cos
3
θ
+
cos
5
θ
16
{\displaystyle \cos ^{5}\theta ={\frac {10\cos \theta +5\cos 3\theta +\cos 5\theta }{16}}}
sin
5
θ
cos
5
θ
=
10
sin
2
θ
−
5
sin
6
θ
+
sin
10
θ
512
{\displaystyle \sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin 2\theta -5\sin 6\theta +\sin 10\theta }{512}}}
Samband för två vinklar
sin
(
x
±
y
)
=
sin
(
x
)
cos
(
y
)
±
cos
(
x
)
sin
(
y
)
cos
(
x
±
y
)
=
cos
(
x
)
cos
(
y
)
∓
sin
(
x
)
sin
(
y
)
tan
(
x
±
y
)
=
tan
(
x
)
±
tan
(
y
)
1
∓
tan
(
x
)
tan
(
y
)
cot
(
x
±
y
)
=
cot
(
x
)
cot
(
y
)
∓
1
cot
(
y
)
±
cot
(
x
)
{\displaystyle {\begin{aligned}\sin(x\pm y)&=\sin(x)\cos(y)\pm \cos(x)\sin(y)\\\cos(x\pm y)&=\cos(x)\cos(y)\mp \sin(x)\sin(y)\\\tan(x\pm y)&={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}\\\cot(x\pm y)&={\frac {\cot(x)\cot(y)\mp 1}{\cot(y)\pm \cot(x)}}\\\end{aligned}}}
Observera att
±
{\displaystyle \pm }
∓
{\displaystyle \mp }
x + y ) = cos(x )cos(y ) - sin(x )sin(y ) medan cos(x - y ) = cos(x )cos(y ) + sin(x )sin(y ).
sin
(
x
)
−
sin
(
y
)
sin
(
x
)
+
sin
(
y
)
=
tan
x
−
y
2
tan
x
+
y
2
cos
(
x
)
−
cos
(
y
)
cos
(
x
)
+
cos
(
y
)
=
−
tan
(
x
+
y
2
)
tan
(
x
−
y
2
)
tan
(
x
)
−
tan
(
y
)
tan
(
x
)
+
tan
(
y
)
=
sin
(
x
−
y
)
sin
(
x
+
y
)
cot
(
x
)
−
cot
(
y
)
cot
(
x
)
+
cot
(
y
)
=
−
sin
(
x
−
y
)
sin
(
x
+
y
)
{\displaystyle {\begin{aligned}{\frac {\sin(x)-\sin(y)}{\sin(x)+\sin(y)}}&={\frac {\tan {\frac {x-y}{2}}}{\tan {\frac {x+y}{2}}}}\\{\frac {\cos(x)-\cos(y)}{\cos(x)+\cos(y)}}&=-\tan \left({\frac {x+y}{2}}\right)\tan \left({\frac {x-y}{2}}\right)\\{\frac {\tan(x)-\tan(y)}{\tan(x)+\tan(y)}}&={\frac {\sin(x-y)}{\sin(x+y)}}\\{\frac {\cot(x)-\cot(y)}{\cot(x)+\cot(y)}}&=-{\frac {\sin(x-y)}{\sin(x+y)}}\\\end{aligned}}}
Summor
sin
(
x
)
+
sin
(
y
)
=
2
sin
(
x
+
y
2
)
cos
(
x
−
y
2
)
cos
(
x
)
+
cos
(
y
)
=
2
cos
(
x
+
y
2
)
cos
(
x
−
y
2
)
tan
(
x
)
+
tan
(
y
)
=
sin
(
x
+
y
)
cos
(
x
)
cos
(
y
)
cot
(
x
)
+
cot
(
y
)
=
sin
(
x
+
y
)
sin
(
x
)
sin
(
y
)
{\displaystyle {\begin{aligned}\sin(x)+\sin(y)&=2\sin \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)\\\cos(x)+\cos(y)&=2\cos \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)\\\tan(x)+\tan(y)&={\frac {\sin(x+y)}{\cos(x)\cos(y)}}\\\cot(x)+\cot(y)&={\frac {\sin(x+y)}{\sin(x)\sin(y)}}\\\end{aligned}}}
sin
(
x
)
−
sin
(
y
)
=
2
cos
(
x
+
y
2
)
sin
(
x
−
y
2
)
cos
(
x
)
−
cos
(
y
)
=
−
2
sin
(
x
+
y
2
)
sin
(
x
−
y
2
)
tan
(
x
)
−
tan
(
y
)
=
sin
(
x
−
y
)
cos
(
x
)
cos
(
y
)
cot
(
x
)
−
cot
(
y
)
=
−
sin
(
x
−
y
)
sin
(
x
)
sin
(
y
)
{\displaystyle {\begin{aligned}\sin(x)-\sin(y)&=2\cos \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)\\\cos(x)-\cos(y)&=-2\sin \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)\\\tan(x)-\tan(y)&={\frac {\sin(x-y)}{\cos(x)\cos(y)}}\\\cot(x)-\cot(y)&=-{\frac {\sin(x-y)}{\sin(x)\sin(y)}}\\\end{aligned}}}
Produkter
sin
(
x
)
sin
(
y
)
=
cos
(
x
−
y
)
−
cos
(
x
+
y
)
2
sin
(
x
)
cos
(
y
)
=
sin
(
x
−
y
)
+
sin
(
x
+
y
)
2
cos
(
x
)
cos
(
y
)
=
cos
(
x
−
y
)
+
cos
(
x
+
y
)
2
{\displaystyle {\begin{aligned}\sin \left(x\right)\sin \left(y\right)&={\cos \left(x-y\right)-\cos \left(x+y\right) \over 2}\\\sin \left(x\right)\cos \left(y\right)&={\sin \left(x-y\right)+\sin \left(x+y\right) \over 2}\\\cos \left(x\right)\cos \left(y\right)&={\cos \left(x-y\right)+\cos \left(x+y\right) \over 2}\\\end{aligned}}}
Inversa funktioner Samband för en vinkel
sin
(
arcsin
(
x
)
)
=
x
cos
(
arccos
(
x
)
)
=
x
tan
(
arctan
(
x
)
)
=
x
cot
(
arccot
(
x
)
)
=
x
sec
(
arcsec
(
x
)
)
=
x
csc
(
arccsc
(
x
)
)
=
x
{\displaystyle {\begin{aligned}\sin(\arcsin(x))&=x&\quad \cos(\arccos(x))&=x\\\tan(\arctan(x))&=x&\quad \cot(\operatorname {arccot}(x))&=x\\\sec(\operatorname {arcsec}(x))&=x&\quad \csc(\operatorname {arccsc}(x))&=x\\\end{aligned}}}
arcsin
(
sin
(
x
)
)
=
x
,
för
−
π
/
2
≤
x
≤
π
/
2
arccos
(
cos
(
x
)
)
=
x
,
för
0
≤
x
≤
π
arctan
(
tan
(
x
)
)
=
x
,
för
−
π
/
2
<
x
<
π
/
2
arccot
(
cot
(
x
)
)
=
x
,
för
0
<
x
<
π
arcsec
(
sec
(
x
)
)
=
x
,
för
0
≤
x
<
π
/
2
eller
π
/
2
<
x
≤
π
arccsc
(
csc
(
x
)
)
=
x
,
för
−
π
/
2
≤
x
<
0
eller
0
<
x
≤
π
/
2
{\displaystyle {\begin{aligned}\arcsin(\sin(x))&=x,{\mbox{ för }}-\pi /2\leq x\leq \pi /2\\\arccos(\cos(x))&=x,{\mbox{ för }}0\leq x\leq \pi \\\arctan(\tan(x))&=x,{\mbox{ för }}-\pi /2<x<\pi /2\\\operatorname {arccot}(\cot(x))&=x,{\mbox{ för }}0<x<\pi \\\operatorname {arcsec}(\sec(x))&=x,{\mbox{ för }}0\leq x<\pi /2{\mbox{ eller }}\pi /2<x\leq \pi \\\operatorname {arccsc}(\csc(x))&=x,{\mbox{ för }}-\pi /2\leq x<0{\mbox{ eller }}0<x\leq \pi /2\\\end{aligned}}}
Kompletterande
arccos
(
x
)
=
π
2
−
arcsin
(
x
)
arccot
(
x
)
=
π
2
−
arctan
(
x
)
arccsc
(
x
)
=
π
2
−
arcsec
(
x
)
{\displaystyle {\begin{aligned}\arccos(x)&={\frac {\pi }{2}}-\arcsin(x)\\\operatorname {arccot}(x)&={\frac {\pi }{2}}-\arctan(x)\\\operatorname {arccsc}(x)&={\frac {\pi }{2}}-\operatorname {arcsec}(x)\\\end{aligned}}}
Likheter för negativa argument
arcsin
(
−
x
)
=
−
arcsin
(
x
)
arccos
(
−
x
)
=
π
−
arccos
(
x
)
arctan
(
−
x
)
=
−
arctan
(
x
)
arccot
(
−
x
)
=
π
−
arccot
(
x
)
arcsec
(
−
x
)
=
π
−
arcsec
(
x
)
arccsc
(
−
x
)
=
−
arccsc
(
x
)
{\displaystyle {\begin{aligned}\arcsin(-x)&=-\arcsin(x)\\\arccos(-x)&=\pi -\arccos(x)\\\arctan(-x)&=-\arctan(x)\\\operatorname {arccot}(-x)&=\pi -\operatorname {arccot}(x)\\\operatorname {arcsec}(-x)&=\pi -\operatorname {arcsec}(x)\\\operatorname {arccsc}(-x)&=-\operatorname {arccsc}(x)\\\end{aligned}}}
Reciproka funktioner
arccos
1
x
=
arcsec
(
x
)
arcsin
1
x
=
arccsc
(
x
)
arctan
1
x
=
π
2
−
arctan
(
x
)
=
arccot
(
x
)
,
om
x
>
0
arctan
1
x
=
−
π
2
−
arctan
(
x
)
=
−
π
+
arccot
(
x
)
,
om
x
<
0
arccot
1
x
=
π
2
−
arccot
(
x
)
=
arctan
(
x
)
,
om
x
>
0
arccot
1
x
=
3
π
2
−
arccot
(
x
)
=
π
+
arctan
(
x
)
,
om
x
<
0
arcsec
1
x
=
arccos
(
x
)
arccsc
1
x
=
arcsin
(
x
)
{\displaystyle {\begin{aligned}\arccos {\frac {1}{x}}&=\operatorname {arcsec}(x)\\\arcsin {\frac {1}{x}}&=\operatorname {arccsc}(x)\\\arctan {\frac {1}{x}}&={\frac {\pi }{2}}-\arctan(x)=\operatorname {arccot}(x),{\text{ om }}x>0\\\arctan {\frac {1}{x}}&=-{\frac {\pi }{2}}-\arctan(x)=-\pi +\operatorname {arccot}(x),{\text{ om }}x<0\\\operatorname {arccot} {\frac {1}{x}}&={\frac {\pi }{2}}-\operatorname {arccot}(x)=\arctan(x),{\text{ om }}x>0\\\operatorname {arccot} {\frac {1}{x}}&={\frac {3\pi }{2}}-\operatorname {arccot}(x)=\pi +\arctan(x),{\text{ om }}x<0\\\operatorname {arcsec} {\frac {1}{x}}&=\arccos(x)\\\operatorname {arccsc} {\frac {1}{x}}&=\arcsin(x)\\\end{aligned}}}
Samband för två vinklar
arcsin
α
±
arcsin
β
=
arcsin
(
α
1
−
β
2
±
β
1
−
α
2
)
arccos
α
±
arccos
β
=
arccos
(
α
β
∓
(
1
−
α
2
)
(
1
−
β
2
)
)
arctan
α
±
arctan
β
=
arctan
(
α
±
β
1
∓
α
β
)
{\displaystyle {\begin{aligned}\arcsin \alpha \pm \arcsin \beta &=\arcsin \left(\alpha {\sqrt {1-\beta ^{2}}}\pm \beta {\sqrt {1-\alpha ^{2}}}\right)\\\arccos \alpha \pm \arccos \beta &=\arccos \left(\alpha \beta \mp {\sqrt {(1-\alpha ^{2})(1-\beta ^{2})}}\right)\\\arctan \alpha \pm \arctan \beta &=\arctan \left({\frac {\alpha \pm \beta }{1\mp \alpha \beta }}\right)\end{aligned}}}
Se även 
