List of trigonometric identities

sin ⁡ θ = {\displaystyle \sin \theta =}

sin ⁡ θ {\displaystyle \sin \theta }

1 csc ⁡ θ {\displaystyle {\frac {1}{\csc \theta }}}

± 1 − cos 2 ⁡ θ {\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}

± sec 2 ⁡ θ − 1 sec ⁡ θ {\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}

± tan ⁡ θ 1 + tan 2 ⁡ θ {\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}

± 1 1 + cot 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}

csc ⁡ θ = {\displaystyle \csc \theta =}

1 sin ⁡ θ {\displaystyle {\frac {1}{\sin \theta }}}

csc ⁡ θ {\displaystyle \csc \theta }

± 1 1 − cos 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}

± sec ⁡ θ sec 2 ⁡ θ − 1 {\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}}

± 1 + tan 2 ⁡ θ tan ⁡ θ {\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}}

± 1 + cot 2 ⁡ θ {\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}

cos ⁡ θ = {\displaystyle \cos \theta =}

± 1 − sin 2 ⁡ θ {\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}

± csc 2 ⁡ θ − 1 csc ⁡ θ {\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}

cos ⁡ θ {\displaystyle \cos \theta }

1 sec ⁡ θ {\displaystyle {\frac {1}{\sec \theta }}}

± 1 1 + tan 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}

± cot ⁡ θ 1 + cot 2 ⁡ θ {\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}

sec ⁡ θ = {\displaystyle \sec \theta =}

± 1 1 − sin 2 ⁡ θ {\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}

± csc ⁡ θ csc 2 ⁡ θ − 1 {\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}}

1 cos ⁡ θ {\displaystyle {\frac {1}{\cos \theta }}}

sec ⁡ θ {\displaystyle \sec \theta }

± 1 + tan 2 ⁡ θ {\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}

± 1 + cot 2 ⁡ θ cot ⁡ θ {\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}}

tan ⁡ θ = {\displaystyle \tan \theta =}

± sin ⁡ θ 1 − sin 2 ⁡ θ {\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}

± 1 csc 2 ⁡ θ − 1 {\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}

± 1 − cos 2 ⁡ θ cos ⁡ θ {\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}

± sec 2 ⁡ θ − 1 {\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}

tan ⁡ θ {\displaystyle \tan \theta }

1 cot ⁡ θ {\displaystyle {\frac {1}{\cot \theta }}}

cot ⁡ θ = {\displaystyle \cot \theta =}

± 1 − sin 2 ⁡ θ sin ⁡ θ {\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}

± csc 2 ⁡ θ − 1 {\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}

± cos ⁡ θ 1 − cos 2 ⁡ θ {\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}}

± 1 sec 2 ⁡ θ − 1 {\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}

1 tan ⁡ θ {\displaystyle {\frac {1}{\tan \theta }}}

cot ⁡ θ {\displaystyle \cot \theta }

sin ⁡ ( − θ ) = − sin ⁡ θ {\displaystyle \sin(-\theta )=-\sin \theta }

sin ⁡ ( π 2 − θ ) = cos ⁡ θ {\displaystyle \sin \left({\tfrac {\pi }{2}}-\theta \right)=\cos \theta }

sin ⁡ ( π − θ ) = + sin ⁡ θ {\displaystyle \sin(\pi -\theta )=+\sin \theta }

sin ⁡ ( 3 π 2 − θ ) = − cos ⁡ θ {\displaystyle \sin \left({\tfrac {3\pi }{2}}-\theta \right)=-\cos \theta }

sin ⁡ ( 2 π − θ ) = − sin ⁡ ( θ ) = sin ⁡ ( − θ ) {\displaystyle \sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )}

cos ⁡ ( − θ ) = + cos ⁡ θ {\displaystyle \cos(-\theta )=+\cos \theta }

cos ⁡ ( π 2 − θ ) = sin ⁡ θ {\displaystyle \cos \left({\tfrac {\pi }{2}}-\theta \right)=\sin \theta }

cos ⁡ ( π − θ ) = − cos ⁡ θ {\displaystyle \cos(\pi -\theta )=-\cos \theta }

cos ⁡ ( 3 π 2 − θ ) = − sin ⁡ θ {\displaystyle \cos \left({\tfrac {3\pi }{2}}-\theta \right)=-\sin \theta }

cos ⁡ ( 2 π − θ ) = + cos ⁡ ( θ ) = cos ⁡ ( − θ ) {\displaystyle \cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )}

tan ⁡ ( − θ ) = − tan ⁡ θ {\displaystyle \tan(-\theta )=-\tan \theta }

tan ⁡ ( π 2 − θ ) = cot ⁡ θ {\displaystyle \tan \left({\tfrac {\pi }{2}}-\theta \right)=\cot \theta }

tan ⁡ ( π − θ ) = − tan ⁡ θ {\displaystyle \tan(\pi -\theta )=-\tan \theta }

tan ⁡ ( 3 π 2 − θ ) = + cot ⁡ θ {\displaystyle \tan \left({\tfrac {3\pi }{2}}-\theta \right)=+\cot \theta }

tan ⁡ ( 2 π − θ ) = − tan ⁡ ( θ ) = tan ⁡ ( − θ ) {\displaystyle \tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )}

csc ⁡ ( − θ ) = − csc ⁡ θ {\displaystyle \csc(-\theta )=-\csc \theta }

csc ⁡ ( π 2 − θ ) = sec ⁡ θ {\displaystyle \csc \left({\tfrac {\pi }{2}}-\theta \right)=\sec \theta }

csc ⁡ ( π − θ ) = + csc ⁡ θ {\displaystyle \csc(\pi -\theta )=+\csc \theta }

csc ⁡ ( 3 π 2 − θ ) = − sec ⁡ θ {\displaystyle \csc \left({\tfrac {3\pi }{2}}-\theta \right)=-\sec \theta }

csc ⁡ ( 2 π − θ ) = − csc ⁡ ( θ ) = csc ⁡ ( − θ ) {\displaystyle \csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )}

sec ⁡ ( − θ ) = + sec ⁡ θ {\displaystyle \sec(-\theta )=+\sec \theta }

sec ⁡ ( π 2 − θ ) = csc ⁡ θ {\displaystyle \sec \left({\tfrac {\pi }{2}}-\theta \right)=\csc \theta }

sec ⁡ ( π − θ ) = − sec ⁡ θ {\displaystyle \sec(\pi -\theta )=-\sec \theta }

sec ⁡ ( 3 π 2 − θ ) = − csc ⁡ θ {\displaystyle \sec \left({\tfrac {3\pi }{2}}-\theta \right)=-\csc \theta }

sec ⁡ ( 2 π − θ ) = + sec ⁡ ( θ ) = sec ⁡ ( − θ ) {\displaystyle \sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )}

cot ⁡ ( − θ ) = − cot ⁡ θ {\displaystyle \cot(-\theta )=-\cot \theta }

cot ⁡ ( π 2 − θ ) = tan ⁡ θ {\displaystyle \cot \left({\tfrac {\pi }{2}}-\theta \right)=\tan \theta }

cot ⁡ ( π − θ ) = − cot ⁡ θ {\displaystyle \cot(\pi -\theta )=-\cot \theta }

cot ⁡ ( 3 π 2 − θ ) = + tan ⁡ θ {\displaystyle \cot \left({\tfrac {3\pi }{2}}-\theta \right)=+\tan \theta }

cot ⁡ ( 2 π − θ ) = − cot ⁡ ( θ ) = cot ⁡ ( − θ ) {\displaystyle \cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )}

sin ⁡ ( θ ± π 2 ) = ± cos ⁡ θ {\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }

sin ⁡ ( θ + π ) = − sin ⁡ θ {\displaystyle \sin(\theta +\pi )=-\sin \theta }

sin ⁡ ( θ + k ⋅ 2 π ) = + sin ⁡ θ {\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }

2 π {\displaystyle 2\pi }

cos ⁡ ( θ ± π 2 ) = ∓ sin ⁡ θ {\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }

cos ⁡ ( θ + π ) = − cos ⁡ θ {\displaystyle \cos(\theta +\pi )=-\cos \theta }

cos ⁡ ( θ + k ⋅ 2 π ) = + cos ⁡ θ {\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }

2 π {\displaystyle 2\pi }

csc ⁡ ( θ ± π 2 ) = ± sec ⁡ θ {\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }

csc ⁡ ( θ + π ) = − csc ⁡ θ {\displaystyle \csc(\theta +\pi )=-\csc \theta }

csc ⁡ ( θ + k ⋅ 2 π ) = + csc ⁡ θ {\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }

2 π {\displaystyle 2\pi }

sec ⁡ ( θ ± π 2 ) = ∓ csc ⁡ θ {\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }

sec ⁡ ( θ + π ) = − sec ⁡ θ {\displaystyle \sec(\theta +\pi )=-\sec \theta }

sec ⁡ ( θ + k ⋅ 2 π ) = + sec ⁡ θ {\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }

2 π {\displaystyle 2\pi }

tan ⁡ ( θ ± π 4 ) = tan ⁡ θ ± 1 1 ∓ tan ⁡ θ {\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}}

tan ⁡ ( θ + π 2 ) = − cot ⁡ θ {\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }

tan ⁡ ( θ + k ⋅ π ) = + tan ⁡ θ {\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }

π {\displaystyle \pi }

cot ⁡ ( θ ± π 4 ) = cot ⁡ θ ∓ 1 1 ± cot ⁡ θ {\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}}

cot ⁡ ( θ + π 2 ) = − tan ⁡ θ {\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }

cot ⁡ ( θ + k ⋅ π ) = + cot ⁡ θ {\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }

π {\displaystyle \pi }

Cosine

cos ⁡ ( α ± β ) {\displaystyle \cos(\alpha \pm \beta )}

= {\displaystyle =}

cos ⁡ α cos ⁡ β ∓ sin ⁡ α sin ⁡ β {\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }

Tangent

tan ⁡ ( α ± β ) {\displaystyle \tan(\alpha \pm \beta )}

= {\displaystyle =}

tan ⁡ α ± tan ⁡ β 1 ∓ tan ⁡ α tan ⁡ β {\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}

Cosecant

csc ⁡ ( α ± β ) {\displaystyle \csc(\alpha \pm \beta )}

= {\displaystyle =}

sec ⁡ α sec ⁡ β csc ⁡ α csc ⁡ β sec ⁡ α csc ⁡ β ± csc ⁡ α sec ⁡ β {\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}

Secant

sec ⁡ ( α ± β ) {\displaystyle \sec(\alpha \pm \beta )}

= {\displaystyle =}

sec ⁡ α sec ⁡ β csc ⁡ α csc ⁡ β csc ⁡ α csc ⁡ β ∓ sec ⁡ α sec ⁡ β {\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}

Cotangent

cot ⁡ ( α ± β ) {\displaystyle \cot(\alpha \pm \beta )}

= {\displaystyle =}

cot ⁡ α cot ⁡ β ∓ 1 cot ⁡ β ± cot ⁡ α {\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}

Arcsine

arcsin ⁡ x ± arcsin ⁡ y {\displaystyle \arcsin x\pm \arcsin y}

= {\displaystyle =}

arcsin ⁡ ( x 1 − y 2 ± y 1 − x 2 y ) {\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}{\vphantom {y}}}}\right)}

Arccosine

arccos ⁡ x ± arccos ⁡ y {\displaystyle \arccos x\pm \arccos y}

= {\displaystyle =}

arccos ⁡ ( x y ∓ ( 1 − x 2 ) ( 1 − y 2 ) ) {\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}

Arctangent

arctan ⁡ x ± arctan ⁡ y {\displaystyle \arctan x\pm \arctan y}

= {\displaystyle =}

arctan ⁡ ( x ± y 1 ∓ x y ) {\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}

Arccotangent

arccot ⁡ x ± arccot ⁡ y {\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}

= {\displaystyle =}

arccot ⁡ ( x y ∓ 1 y ± x ) {\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}

de Moivre's formula, i is the imaginary unit

cos ⁡ ( n θ ) + i sin ⁡ ( n θ ) = ( cos ⁡ θ + i sin ⁡ θ ) n {\displaystyle \cos(n\theta )+i\sin(n\theta )=(\cos \theta +i\sin \theta )^{n}}

Double-angle formula

sin ⁡ ( 2 θ ) = 2 sin ⁡ θ cos ⁡ θ   = 2 tan ⁡ θ 1 + tan 2 ⁡ θ {\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}}

cos ⁡ ( 2 θ ) = cos 2 ⁡ θ − sin 2 ⁡ θ = 2 cos 2 ⁡ θ − 1 = 1 − 2 sin 2 ⁡ θ = 1 − tan 2 ⁡ θ 1 + tan 2 ⁡ θ {\displaystyle {\begin{aligned}\cos(2\theta )&=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}}

tan ⁡ ( 2 θ ) = 2 tan ⁡ θ 1 − tan 2 ⁡ θ {\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}

cot ⁡ ( 2 θ ) = cot 2 ⁡ θ − 1 2 cot ⁡ θ {\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}}

Triple-angle formula

sin ⁡ ( 3 θ ) = − sin 3 ⁡ θ + 3 cos 2 ⁡ θ sin ⁡ θ = − 4 sin 3 ⁡ θ + 3 sin ⁡ θ {\displaystyle {\begin{aligned}\sin(3\theta )&=-\sin ^{3}\theta +3\cos ^{2}\theta \sin \theta \\&=-4\sin ^{3}\theta +3\sin \theta \end{aligned}}}

cos ⁡ ( 3 θ ) = cos 3 ⁡ θ − 3 sin 2 ⁡ θ cos ⁡ θ = 4 cos 3 ⁡ θ − 3 cos ⁡ θ {\displaystyle {\begin{aligned}\cos(3\theta )&=\cos ^{3}\theta -3\sin ^{2}\theta \cos \theta \\&=4\cos ^{3}\theta -3\cos \theta \end{aligned}}}

tan ⁡ ( 3 θ ) = 3 tan ⁡ θ − tan 3 ⁡ θ 1 − 3 tan 2 ⁡ θ {\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}}

cot ⁡ ( 3 θ ) = 3 cot ⁡ θ − cot 3 ⁡ θ 1 − 3 cot 2 ⁡ θ {\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta }}}

Half-angle formula

sin ⁡ θ 2 = sgn ⁡ ( sin ⁡ θ 2 ) 1 − cos ⁡ θ 2 ( or  sin 2 ⁡ θ 2 = 1 − cos ⁡ θ 2 ) {\displaystyle {\begin{aligned}&\sin {\frac {\theta }{2}}=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\\\&\left({\text{or }}\sin ^{2}{\frac {\theta }{2}}={\frac {1-\cos \theta }{2}}\right)\end{aligned}}}

cos ⁡ θ 2 = sgn ⁡ ( cos ⁡ θ 2 ) 1 + cos ⁡ θ 2 ( or  cos 2 ⁡ θ 2 = 1 + cos ⁡ θ 2 ) {\displaystyle {\begin{aligned}&\cos {\frac {\theta }{2}}=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\\\&\left({\text{or }}\cos ^{2}{\frac {\theta }{2}}={\frac {1+\cos \theta }{2}}\right)\end{aligned}}}

tan ⁡ θ 2 = csc ⁡ θ − cot ⁡ θ = ± 1 − cos ⁡ θ 1 + cos ⁡ θ = sin ⁡ θ 1 + cos ⁡ θ = 1 − cos ⁡ θ sin ⁡ θ tan ⁡ η + θ 2 = sin ⁡ η + sin ⁡ θ cos ⁡ η + cos ⁡ θ tan ⁡ ( θ 2 + π 4 ) = sec ⁡ θ + tan ⁡ θ 1 − sin ⁡ θ 1 + sin ⁡ θ = | 1 − tan ⁡ θ 2 | | 1 + tan ⁡ θ 2 | tan ⁡ θ 2 = tan ⁡ θ 1 + 1 + tan 2 ⁡ θ for  θ ∈ ( − π 2 , π 2 ) {\displaystyle {\begin{aligned}\tan {\frac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1+\cos \theta }}\\[3pt]&={\frac {1-\cos \theta }{\sin \theta }}\\[5pt]\tan {\frac {\eta +\theta }{2}}&={\frac {\sin \eta +\sin \theta }{\cos \eta +\cos \theta }}\\[5pt]\tan \left({\frac {\theta }{2}}+{\frac {\pi }{4}}\right)&=\sec \theta +\tan \theta \\[5pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}&={\frac {\left|1-\tan {\frac {\theta }{2}}\right|}{\left|1+\tan {\frac {\theta }{2}}\right|}}\\[5pt]\tan {\frac {\theta }{2}}&={\frac {\tan \theta }{1+{\sqrt {1+\tan ^{2}\theta }}}}\\&{\text{for }}\theta \in \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)\end{aligned}}}

cot ⁡ θ 2 = csc ⁡ θ + cot ⁡ θ = ± 1 + cos ⁡ θ 1 − cos ⁡ θ = sin ⁡ θ 1 − cos ⁡ θ = 1 + cos ⁡ θ sin ⁡ θ {\displaystyle {\begin{aligned}\cot {\frac {\theta }{2}}&=\csc \theta +\cot \theta \\&=\pm \,{\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\[3pt]&={\frac {\sin \theta }{1-\cos \theta }}\\[4pt]&={\frac {1+\cos \theta }{\sin \theta }}\end{aligned}}}

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}}}

n is odd

cos n ⁡ θ = 2 2 n ∑ k = 0 n − 1 2 ( n k ) cos ⁡ ( ( n − 2 k ) θ ) {\displaystyle \cos ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}

sin n ⁡ θ = 2 2 n ∑ k = 0 n − 1 2 ( − 1 ) ( n − 1 2 − k ) ( n k ) sin ⁡ ( ( n − 2 k ) θ ) {\displaystyle \sin ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}(-1)^{\left({\frac {n-1}{2}}-k\right)}{\binom {n}{k}}\sin {{\big (}(n-2k)\theta {\big )}}}

n is even

cos n ⁡ θ = 1 2 n ( n n 2 ) + 2 2 n ∑ k = 0 n 2 − 1 ( n k ) cos ⁡ ( ( n − 2 k ) θ ) {\displaystyle \cos ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}

sin n ⁡ θ = 1 2 n ( n n 2 ) + 2 2 n ∑ k = 0 n 2 − 1 ( − 1 ) ( n 2 − k ) ( n k ) cos ⁡ ( ( n − 2 k ) θ ) {\displaystyle \sin ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{\left({\frac {n}{2}}-k\right)}{\binom {n}{k}}\cos {{\big (}(n-2k)\theta {\big )}}}

sin ⁡ θ = e i θ − e − i θ 2 i {\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}

arcsin ⁡ x = − i ln ⁡ ( i x + 1 − x 2 ) {\displaystyle \arcsin x=-i\,\ln \left(ix+{\sqrt {1-x^{2}}}\right)}

cos ⁡ θ = e i θ + e − i θ 2 {\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}

arccos ⁡ x = − i ln ⁡ ( x + x 2 − 1 ) {\displaystyle \arccos x=-i\ln \left(x+{\sqrt {x^{2}-1}}\right)}

tan ⁡ θ = − i e i θ − e − i θ e i θ + e − i θ {\displaystyle \tan \theta =-i\,{\frac {e^{i\theta }-e^{-i\theta }}{e^{i\theta }+e^{-i\theta }}}}

arctan ⁡ x = i 2 ln ⁡ ( i + x i − x ) {\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)}

csc ⁡ θ = 2 i e i θ − e − i θ {\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}}

arccsc ⁡ x = − i ln ⁡ ( i x + 1 − 1 x 2 ) {\displaystyle \operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}

sec ⁡ θ = 2 e i θ + e − i θ {\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}}

arcsec ⁡ x = − i ln ⁡ ( 1 x + i 1 − 1 x 2 ) {\displaystyle \operatorname {arcsec} x=-i\,\ln \left({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}

cot ⁡ θ = i e i θ + e − i θ e i θ − e − i θ {\displaystyle \cot \theta =i\,{\frac {e^{i\theta }+e^{-i\theta }}{e^{i\theta }-e^{-i\theta }}}}

arccot ⁡ x = i 2 ln ⁡ ( x − i x + i ) {\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)}

cis ⁡ θ = e i θ {\displaystyle \operatorname {cis} \theta =e^{i\theta }}

arccis ⁡ x = − i ln ⁡ x {\displaystyle \operatorname {arccis} x=-i\ln x}