In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Pythagorean identities

 Main article: Pythagorean trigonometric identity

The basic relationship between the sine and cosine is given by the Pythagorean identity:

${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}$

where ${\displaystyle \sin ^{2}\theta }$ means ${\displaystyle (\sin \theta )^{2))$ and ${\displaystyle \cos ^{2}\theta }$ means ${\displaystyle (\cos \theta )^{2}.}$

This can be viewed as a version of the Pythagorean theorem, and follows from the equation ${\displaystyle x^{2}+y^{2}=1}$ for the unit circle. This equation can be solved for either the sine or the cosine:

{\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta )),\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta )).\end{aligned))}

where the sign depends on the quadrant of ${\displaystyle \theta .}$

Dividing this identity by ${\displaystyle \sin ^{2}\theta }$, ${\displaystyle \cos ^{2}\theta }$, or both yields the following identities:

{\displaystyle {\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \\&1+\tan ^{2}\theta =\sec ^{2}\theta \\&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned))}

Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):

Each trigonometric function in terms of each of the other five.[1]
in terms of ${\displaystyle \sin \theta }$ ${\displaystyle \csc \theta }$ ${\displaystyle \cos \theta }$ ${\displaystyle \sec \theta }$ ${\displaystyle \tan \theta }$ ${\displaystyle \cot \theta }$
${\displaystyle \sin \theta =}$ ${\displaystyle \sin \theta }$ ${\displaystyle {\frac {1}{\csc \theta ))}$ ${\displaystyle \pm {\sqrt {1-\cos ^{2}\theta ))}$ ${\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1)){\sec \theta ))}$ ${\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta ))))$ ${\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta ))))$
${\displaystyle \csc \theta =}$ ${\displaystyle {\frac {1}{\sin \theta ))}$ ${\displaystyle \csc \theta }$ ${\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta ))))$ ${\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1))))$ ${\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta )){\tan \theta ))}$ ${\displaystyle \pm {\sqrt {1+\cot ^{2}\theta ))}$
${\displaystyle \cos \theta =}$ ${\displaystyle \pm {\sqrt {1-\sin ^{2}\theta ))}$ ${\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1)){\csc \theta ))}$ ${\displaystyle \cos \theta }$ ${\displaystyle {\frac {1}{\sec \theta ))}$ ${\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta ))))$ ${\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta ))))$
${\displaystyle \sec \theta =}$ ${\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta ))))$ ${\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1))))$ ${\displaystyle {\frac {1}{\cos \theta ))}$ ${\displaystyle \sec \theta }$ ${\displaystyle \pm {\sqrt {1+\tan ^{2}\theta ))}$ ${\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta )){\cot \theta ))}$
${\displaystyle \tan \theta =}$ ${\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta ))))$ ${\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1))))$ ${\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta )){\cos \theta ))}$ ${\displaystyle \pm {\sqrt {\sec ^{2}\theta -1))}$ ${\displaystyle \tan \theta }$ ${\displaystyle {\frac {1}{\cot \theta ))}$
${\displaystyle \cot \theta =}$ ${\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta )){\sin \theta ))}$ ${\displaystyle \pm {\sqrt {\csc ^{2}\theta -1))}$ ${\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta ))))$ ${\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1))))$ ${\displaystyle {\frac {1}{\tan \theta ))}$ ${\displaystyle \cot \theta }$

Reflections, shifts, and periodicity

By examining the unit circle, one can establish the following properties of the trigonometric functions.

Reflections

When the direction of a Euclidean vector is represented by an angle ${\displaystyle \theta ,}$ this is the angle determined by the free vector (starting at the origin) and the positive ${\displaystyle x}$-unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive ${\displaystyle x}$-axis. If a line (vector) with direction ${\displaystyle \theta }$ is reflected about a line with direction ${\displaystyle \alpha ,}$ then the direction angle ${\displaystyle \theta ^{\prime ))$ of this reflected line (vector) has the value

${\displaystyle \theta ^{\prime }=2\alpha -\theta .}$

The values of the trigonometric functions of these angles ${\displaystyle \theta ,\;\theta ^{\prime ))$ for specific angles ${\displaystyle \alpha }$ satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as reduction formulae.[2]

${\displaystyle \theta }$ reflected in ${\displaystyle \alpha =0}$[3]
odd/even identities
${\displaystyle \theta }$ reflected in ${\displaystyle \alpha ={\frac {\pi }{4))}$ ${\displaystyle \theta }$ reflected in ${\displaystyle \alpha ={\frac {\pi }{2))}$ ${\displaystyle \theta }$ reflected in ${\displaystyle \alpha ={\frac {3\pi }{4))}$ ${\displaystyle \theta }$ reflected in ${\displaystyle \alpha =\pi }$
compare to ${\displaystyle \alpha =0}$
${\displaystyle \sin(-\theta )=-\sin \theta }$ ${\displaystyle \sin \left({\tfrac {\pi }{2))-\theta \right)=\cos \theta }$ ${\displaystyle \sin(\pi -\theta )=+\sin \theta }$ ${\displaystyle \sin \left({\tfrac {3\pi }{2))-\theta \right)=-\cos \theta }$ ${\displaystyle \sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )}$
${\displaystyle \cos(-\theta )=+\cos \theta }$ ${\displaystyle \cos \left({\tfrac {\pi }{2))-\theta \right)=\sin \theta }$ ${\displaystyle \cos(\pi -\theta )=-\cos \theta }$ ${\displaystyle \cos \left({\tfrac {3\pi }{2))-\theta \right)=-\sin \theta }$ ${\displaystyle \cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )}$
${\displaystyle \tan(-\theta )=-\tan \theta }$ ${\displaystyle \tan \left({\tfrac {\pi }{2))-\theta \right)=\cot \theta }$ ${\displaystyle \tan(\pi -\theta )=-\tan \theta }$ ${\displaystyle \tan \left({\tfrac {3\pi }{2))-\theta \right)=+\cot \theta }$ ${\displaystyle \tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )}$
${\displaystyle \csc(-\theta )=-\csc \theta }$ ${\displaystyle \csc \left({\tfrac {\pi }{2))-\theta \right)=\sec \theta }$ ${\displaystyle \csc(\pi -\theta )=+\csc \theta }$ ${\displaystyle \csc \left({\tfrac {3\pi }{2))-\theta \right)=-\sec \theta }$ ${\displaystyle \csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )}$
${\displaystyle \sec(-\theta )=+\sec \theta }$ ${\displaystyle \sec \left({\tfrac {\pi }{2))-\theta \right)=\csc \theta }$ ${\displaystyle \sec(\pi -\theta )=-\sec \theta }$ ${\displaystyle \sec \left({\tfrac {3\pi }{2))-\theta \right)=-\csc \theta }$ ${\displaystyle \sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )}$
${\displaystyle \cot(-\theta )=-\cot \theta }$ ${\displaystyle \cot \left({\tfrac {\pi }{2))-\theta \right)=\tan \theta }$ ${\displaystyle \cot(\pi -\theta )=-\cot \theta }$ ${\displaystyle \cot \left({\tfrac {3\pi }{2))-\theta \right)=+\tan \theta }$ ${\displaystyle \cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )}$

Shifts and periodicity

Shift by one quarter period Shift by one half period Shift by full periods[4] Period
${\displaystyle \sin(\theta \pm {\tfrac {\pi }{2)))=\pm \cos \theta }$ ${\displaystyle \sin(\theta +\pi )=-\sin \theta }$ ${\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }$ ${\displaystyle 2\pi }$
${\displaystyle \cos(\theta \pm {\tfrac {\pi }{2)))=\mp \sin \theta }$ ${\displaystyle \cos(\theta +\pi )=-\cos \theta }$ ${\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }$ ${\displaystyle 2\pi }$
${\displaystyle \csc(\theta \pm {\tfrac {\pi }{2)))=\pm \sec \theta }$ ${\displaystyle \csc(\theta +\pi )=-\csc \theta }$ ${\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }$ ${\displaystyle 2\pi }$
${\displaystyle \sec(\theta \pm {\tfrac {\pi }{2)))=\mp \csc \theta }$ ${\displaystyle \sec(\theta +\pi )=-\sec \theta }$ ${\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }$ ${\displaystyle 2\pi }$
${\displaystyle \tan(\theta \pm {\tfrac {\pi }{4)))={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta ))}$ ${\displaystyle \tan(\theta +{\tfrac {\pi }{2)))=-\cot \theta }$ ${\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }$ ${\displaystyle \pi }$
${\displaystyle \cot(\theta \pm {\tfrac {\pi }{4)))={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta ))}$ ${\displaystyle \cot(\theta +{\tfrac {\pi }{2)))=-\tan \theta }$ ${\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }$ ${\displaystyle \pi }$

Signs

The sign of trigonometric functions depends on quadrant of the angle. If ${\displaystyle {-\pi }<\theta \leq \pi }$ and sgn is the sign function,

{\displaystyle {\begin{aligned}\operatorname {sgn}(\sin \theta )=\operatorname {sgn}(\csc \theta )&={\begin{cases}+1&{\text{if))\ \ 0<\theta <\pi \\-1&{\text{if))\ \ {-\pi }<\theta <0\\0&{\text{if))\ \ \theta \in \{0,\pi \}\end{cases))\\[5mu]\operatorname {sgn}(\cos \theta )=\operatorname {sgn}(\sec \theta )&={\begin{cases}+1&{\text{if))\ \ {-{\tfrac {1}{2))\pi }<\theta <{\tfrac {1}{2))\pi \\-1&{\text{if))\ \ {-\pi }<\theta <-{\tfrac {1}{2))\pi \ \ {\text{or))\ \ {\tfrac {1}{2))\pi <\theta <\pi \\0&{\text{if))\ \ \theta \in {\bigl \{}{-{\tfrac {1}{2))\pi },{\tfrac {1}{2))\pi {\bigr \))\end{cases))\\[5mu]\operatorname {sgn}(\tan \theta )=\operatorname {sgn}(\cot \theta )&={\begin{cases}+1&{\text{if))\ \ {-\pi }<\theta <-{\tfrac {1}{2))\pi \ \ {\text{or))\ \ 0<\theta <{\tfrac {1}{2))\pi \\-1&{\text{if))\ \ {-{\tfrac {1}{2))\pi }<\theta <0\ \ {\text{or))\ \ {\tfrac {1}{2))\pi <\theta <\pi \\0&{\text{if))\ \ \theta \in {\bigl \{}{-{\tfrac {1}{2))\pi },0,{\tfrac {1}{2))\pi ,\pi {\bigr \))\end{cases))\end{aligned))}

The trigonometric functions are periodic with common period ${\displaystyle 2\pi ,}$ so for values of θ outside the interval ${\displaystyle ({-\pi },\pi ],}$ they take repeating values (see § Shifts and periodicity above).

Angle sum and difference identities

These are also known as the angle addition and subtraction theorems (or formulae).

{\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned))}

The angle difference identities for ${\displaystyle \sin(\alpha -\beta )}$ and ${\displaystyle \cos(\alpha -\beta )}$ can be derived from the angle sum versions by substituting ${\displaystyle -\beta }$ for ${\displaystyle \beta }$ and using the facts that ${\displaystyle \sin(-\beta )=-\sin(\beta )}$ and ${\displaystyle \cos(-\beta )=\cos(\beta )}$. They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here.

These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.

Sine Cosine Tangent ${\displaystyle \sin(\alpha \pm \beta )}$ ${\displaystyle =}$ ${\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$[5][6] ${\displaystyle \cos(\alpha \pm \beta )}$ ${\displaystyle =}$ ${\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }$[6][7] ${\displaystyle \tan(\alpha \pm \beta )}$ ${\displaystyle =}$ ${\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta ))}$[6][8] ${\displaystyle \csc(\alpha \pm \beta )}$ ${\displaystyle =}$ ${\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta ))}$[9] ${\displaystyle \sec(\alpha \pm \beta )}$ ${\displaystyle =}$ ${\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta ))}$[9] ${\displaystyle \cot(\alpha \pm \beta )}$ ${\displaystyle =}$ ${\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha ))}$[6][10] ${\displaystyle \arcsin x\pm \arcsin y}$ ${\displaystyle =}$ ${\displaystyle \arcsin \left(x{\sqrt {1-y^{2))}\pm y{\sqrt {1-x^{2))}\right)}$[11] ${\displaystyle \arccos x\pm \arccos y}$ ${\displaystyle =}$ ${\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)))\right)}$[12] ${\displaystyle \arctan x\pm \arctan y}$ ${\displaystyle =}$ ${\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy))\right)}$[13] ${\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}$ ${\displaystyle =}$ ${\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x))\right)}$

Sines and cosines of sums of infinitely many angles

When the series ${\textstyle \sum _{i=1}^{\infty }\theta _{i))$ converges absolutely then

{\displaystyle {\begin{aligned}{\sin }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggl )}&=\sum _((\text{odd))\ k\geq 1}(-1)^{\frac {k-1}{2))\!\!\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix)){\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}\\{\cos }{\biggl (}\sum _{i=1}^{\infty }\theta _{i}{\biggr )}&=\sum _((\text{even))\ k\geq 0}(-1)^{\frac {k}{2))\,\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix)){\biggl (}\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}{\biggr )}.\end{aligned))}

Because the series ${\textstyle \sum _{i=1}^{\infty }\theta _{i))$ converges absolutely, it is necessarily the case that ${\textstyle \lim _{i\to \infty }\theta _{i}=0,}$ ${\textstyle \lim _{i\to \infty }\sin \theta _{i}=0,}$ and ${\textstyle \lim _{i\to \infty }\cos \theta _{i}=1.}$ In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.

When only finitely many of the angles ${\displaystyle \theta _{i))$ are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.

Tangents and cotangents of sums

Let ${\displaystyle e_{k))$ (for ${\displaystyle k=0,1,2,3,\ldots }$) be the kth-degree elementary symmetric polynomial in the variables

${\displaystyle x_{i}=\tan \theta _{i))$
for ${\displaystyle i=0,1,2,3,\ldots ,}$ that is,

{\displaystyle {\begin{aligned}e_{0}&=1\\[6pt]e_{1}&=\sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i

Then

{\displaystyle {\begin{aligned}{\tan }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac ((\sin }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i))((\cos }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i))}\\&={\frac {\displaystyle \sum _((\text{odd))\ k\geq 1}(-1)^{\frac {k-1}{2))\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix))\prod _{i\in A}\tan \theta _{i)){\displaystyle \sum _((\text{even))\ k\geq 0}~(-1)^{\frac {k}{2))~~\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix))\prod _{i\in A}\tan \theta _{i))}={\frac {e_{1}-e_{3}+e_{5}-\cdots }{e_{0}-e_{2}+e_{4}-\cdots ))\\{\cot }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {e_{0}-e_{2}+e_{4}-\cdots }{e_{1}-e_{3}+e_{5}-\cdots ))\end{aligned))}

using the sine and cosine sum formulae above.

The number of terms on the right side depends on the number of terms on the left side.

For example:

{\displaystyle {\begin{aligned}\tan(\theta _{1}+\theta _{2})&={\frac {e_{1)){e_{0}-e_{2))}={\frac {x_{1}+x_{2)){1\ -\ x_{1}x_{2))}={\frac {\tan \theta _{1}+\tan \theta _{2)){1\ -\ \tan \theta _{1}\tan \theta _{2))},\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3})&={\frac {e_{1}-e_{3)){e_{0}-e_{2))}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}))),\\[8pt]\tan(\theta _{1}+\theta _{2}+\theta _{3}+\theta _{4})&={\frac {e_{1}-e_{3)){e_{0}-e_{2}+e_{4))}\\[8pt]&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4}))),\end{aligned))}

and so on. The case of only finitely many terms can be proved by mathematical induction.[14]

Secants and cosecants of sums

{\displaystyle {\begin{aligned}{\sec }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {\prod _{i}\sec \theta _{i)){e_{0}-e_{2}+e_{4}-\cdots ))\\[8pt]{\csc }{\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {\prod _{i}\sec \theta _{i)){e_{1}-e_{3}+e_{5}-\cdots ))\end{aligned))}

where ${\displaystyle e_{k))$ is the kth-degree elementary symmetric polynomial in the n variables ${\displaystyle x_{i}=\tan \theta _{i},}$ ${\displaystyle i=1,\ldots ,n,}$ and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left.[15] The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma ))\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma )).\end{aligned))}

Ptolemy's theorem

 Main article: Ptolemy's theorem

Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a cyclic quadrilateral ${\displaystyle ABCD}$, as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities.[16] The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.

By Thales's theorem, ${\displaystyle \angle DAB}$ and ${\displaystyle \angle DCB}$ are both right angles. The right-angled triangles ${\displaystyle DAB}$ and ${\displaystyle DCB}$ both share the hypotenuse ${\displaystyle {\overline {BD))}$ of length 1. Thus, the side ${\displaystyle {\overline {AB))=\sin \alpha }$, ${\displaystyle {\overline {AD))=\cos \alpha }$, ${\displaystyle {\overline {BC))=\sin \beta }$ and ${\displaystyle {\overline {CD))=\cos \beta }$.

By the inscribed angle theorem, the central angle subtended by the chord ${\displaystyle {\overline {AC))}$ at the circle's center is twice the angle ${\displaystyle \angle ADC}$, i.e. ${\displaystyle 2(\alpha +\beta )}$. Therefore, the symmetrical pair of red triangles each has the angle ${\displaystyle \alpha +\beta }$ at the center. Each of these triangles has a hypotenuse of length ${\textstyle {\frac {1}{2))}$, so the length of ${\displaystyle {\overline {AC))}$ is ${\textstyle 2\times {\frac {1}{2))\sin(\alpha +\beta )}$, i.e. simply ${\displaystyle \sin(\alpha +\beta )}$. The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also ${\displaystyle \sin(\alpha +\beta )}$.

When these values are substituted into the statement of Ptolemy's theorem that ${\displaystyle |{\overline {AC))|\cdot |{\overline {BD))|=|{\overline {AB))|\cdot |{\overline {CD))|+|{\overline {AD))|\cdot |{\overline {BC))|}$, this yields the angle sum trigonometric identity for sine: ${\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta }$. The angle difference formula for ${\displaystyle \sin(\alpha -\beta )}$ can be similarly derived by letting the side ${\displaystyle {\overline {CD))}$ serve as a diameter instead of ${\displaystyle {\overline {BD))}$.[16]

Multiple-angle and half-angle formulae

Tn is the nth Chebyshev polynomial ${\displaystyle \cos(n\theta )=T_{n}(\cos \theta )}$[17] ${\displaystyle \cos(n\theta )+i\sin(n\theta )=(\cos \theta +i\sin \theta )^{n))$[18]

Multiple-angle formulae

Double-angle formulae

Formulae for twice an angle.[19]

• ${\displaystyle \sin(2\theta )=2\sin \theta \cos \theta =(\sin \theta +\cos \theta )^{2}-1={\frac {2\tan \theta }{1+\tan ^{2}\theta ))}$
• ${\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta ={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta ))}$
• ${\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta ))}$
• ${\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta ))={\frac {1-\tan ^{2}\theta }{2\tan \theta ))}$
• ${\displaystyle \sec(2\theta )={\frac {\sec ^{2}\theta }{2-\sec ^{2}\theta ))={\frac {1+\tan ^{2}\theta }{1-\tan ^{2}\theta ))}$
• ${\displaystyle \csc(2\theta )={\frac {\sec \theta \csc \theta }{2))={\frac {1+\tan ^{2}\theta }{2\tan \theta ))}$

Triple-angle formulae

Formulae for triple angles.[19]

• ${\displaystyle \sin(3\theta )=3\sin \theta -4\sin ^{3}\theta =4\sin \theta \sin \left({\frac {\pi }{3))-\theta \right)\sin \left({\frac {\pi }{3))+\theta \right)}$
• ${\displaystyle \cos(3\theta )=4\cos ^{3}\theta -3\cos \theta =4\cos \theta \cos \left({\frac {\pi }{3))-\theta \right)\cos \left({\frac {\pi }{3))+\theta \right)}$
• ${\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta ))=\tan \theta \tan \left({\frac {\pi }{3))-\theta \right)\tan \left({\frac {\pi }{3))+\theta \right)}$
• ${\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta ))}$
• ${\displaystyle \sec(3\theta )={\frac {\sec ^{3}\theta }{4-3\sec ^{2}\theta ))}$
• ${\displaystyle \csc(3\theta )={\frac {\csc ^{3}\theta }{3\csc ^{2}\theta -4))}$

Multiple-angle formulae

Formulae for multiple angles.[20]

• {\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ odd))}(-1)^{\frac {k-1}{2)){n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sin \theta \sum _{i=0}^{(n+1)/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i+1}{i \choose j}\cos ^{n-2(i-j)-1}\theta \\{}&=2^{(n-1)}\prod _{k=0}^{n-1}\sin(k\pi /n+\theta )\end{aligned))}
• ${\displaystyle \cos(n\theta )=\sum _{k{\text{ even))}(-1)^{\frac {k}{2)){n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sum _{i=0}^{n/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i}{i \choose j}\cos ^{n-2(i-j)}\theta }$
• ${\displaystyle \cos((2n+1)\theta )=(-1)^{n}2^{2n}\prod _{k=0}^{2n}\cos(k\pi /(2n+1)-\theta )}$
• ${\displaystyle \cos(2n\theta )=(-1)^{n}2^{2n-1}\prod _{k=0}^{2n-1}\cos((1+2k)\pi /(4n)-\theta )}$
• ${\displaystyle \tan(n\theta )={\frac {\sum _{k{\text{ odd))}(-1)^{\frac {k-1}{2)){n \choose k}\tan ^{k}\theta }{\sum _{k{\text{ even))}(-1)^{\frac {k}{2)){n \choose k}\tan ^{k}\theta ))}$

Chebyshev method

The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the ${\displaystyle (n-1)}$th and ${\displaystyle (n-2)}$th values.[21]

${\displaystyle \cos(nx)}$ can be computed from ${\displaystyle \cos((n-1)x)}$, ${\displaystyle \cos((n-2)x)}$, and ${\displaystyle \cos(x)}$ with

${\displaystyle \cos(nx)=2\cos x\cos((n-1)x)-\cos((n-2)x).}$

This can be proved by adding together the formulae

{\displaystyle {\begin{aligned}\cos((n-1)x+x)&=\cos((n-1)x)\cos x-\sin((n-1)x)\sin x\\\cos((n-1)x-x)&=\cos((n-1)x)\cos x+\sin((n-1)x)\sin x\end{aligned))}

It follows by induction that ${\displaystyle \cos(nx)}$ is a polynomial of ${\displaystyle \cos x,}$ the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.

Similarly, ${\displaystyle \sin(nx)}$ can be computed from ${\displaystyle \sin((n-1)x),}$ ${\displaystyle \sin((n-2)x),}$ and ${\displaystyle \cos x}$ with

${\displaystyle \sin(nx)=2\cos x\sin((n-1)x)-\sin((n-2)x)}$
This can be proved by adding formulae for ${\displaystyle \sin((n-1)x+x)}$ and ${\displaystyle \sin((n-1)x-x).}$

Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

${\displaystyle \tan(nx)={\frac {\tan((n-1)x)+\tan x}{1-\tan((n-1)x)\tan x))\,.}$

Half-angle formulae

{\displaystyle {\begin{aligned}\sin {\frac {\theta }{2))&=\operatorname {sgn} \left(\sin {\frac {\theta }{2))\right){\sqrt {\frac {1-\cos \theta }{2))}\\[3pt]\cos {\frac {\theta }{2))&=\operatorname {sgn} \left(\cos {\frac {\theta }{2))\right){\sqrt {\frac {1+\cos \theta }{2))}\\[3pt]\tan {\frac {\theta }{2))&={\frac {1-\cos \theta }{\sin \theta ))={\frac {\sin \theta }{1+\cos \theta ))=\csc \theta -\cot \theta ={\frac {\tan \theta }{1+\sec {\theta ))}\\[6mu]&=\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1-\cos \theta }{1+\cos \theta ))}={\frac {-1+\operatorname {sgn}(\cos \theta ){\sqrt {1+\tan ^{2}\theta ))}{\tan \theta ))\\[3pt]\cot {\frac {\theta }{2))&={\frac {1+\cos \theta }{\sin \theta ))={\frac {\sin \theta }{1-\cos \theta ))=\csc \theta +\cot \theta =\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1+\cos \theta }{1-\cos \theta ))}\\\sec {\frac {\theta }{2))&=\operatorname {sgn} \left(\cos {\frac {\theta }{2))\right){\sqrt {\frac {2}{1+\cos \theta ))}\\\csc {\frac {\theta }{2))&=\operatorname {sgn} \left(\sin {\frac {\theta }{2))\right){\sqrt {\frac {2}{1-\cos \theta ))}\\\end{aligned))}
[22][23]

Also

{\displaystyle {\begin{aligned}\tan {\frac {\eta \pm \theta }{2))&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta ))\\[3pt]\tan \left({\frac {\theta }{2))+{\frac {\pi }{4))\right)&=\sec \theta +\tan \theta \\[3pt]{\sqrt {\frac {1-\sin \theta }{1+\sin \theta ))}&={\frac {\left|1-\tan {\frac {\theta }{2))\right|}{\left|1+\tan {\frac {\theta }{2))\right|))\end{aligned))}

Table

These can be shown by using either the sum and difference identities or the multiple-angle formulae.

Sine Cosine Tangent Cotangent
Double-angle formula[24][25] {\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta ))\end{aligned))} {\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))} ${\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta ))}$ ${\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta ))}$
Triple-angle formula[17][26] {\displaystyle {\begin{aligned}\sin(3\theta )&=-\sin ^{3}\theta +3\cos ^{2}\theta \sin \theta \\&=-4\sin ^{3}\theta +3\sin \theta \end{aligned))} {\displaystyle {\begin{aligned}\cos(3\theta )&=\cos ^{3}\theta -3\sin ^{2}\theta \cos \theta \\&=4\cos ^{3}\theta -3\cos \theta \end{aligned))} ${\displaystyle \tan(3\theta )={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta ))}$ ${\displaystyle \cot(3\theta )={\frac {3\cot \theta -\cot ^{3}\theta }{1-3\cot ^{2}\theta ))}$
Half-angle formula[22][23] {\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))} {\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))} {\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))} {\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))}

The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by field theory.[citation needed]

A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x3 − 3x + d = 0, where ${\displaystyle x}$ is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions is reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

Power-reduction formulae

Obtained by solving the second and third versions of the cosine double-angle formula.

Sine Cosine Other
${\displaystyle \sin ^{2}\theta ={\frac {1-\cos(2\theta )}{2))}$ ${\displaystyle \cos ^{2}\theta ={\frac {1+\cos(2\theta )}{2))}$ ${\displaystyle \sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos(4\theta )}{8))}$
${\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin(3\theta )}{4))}$ ${\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos(3\theta )}{4))}$ ${\displaystyle \sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin(2\theta )-\sin(6\theta )}{32))}$
${\displaystyle \sin ^{4}\theta ={\frac {3-4\cos(2\theta )+\cos(4\theta )}{8))}$ ${\displaystyle \cos ^{4}\theta ={\frac {3+4\cos(2\theta )+\cos(4\theta )}{8))}$ ${\displaystyle \sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos(4\theta )+\cos(8\theta )}{128))}$
${\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin(3\theta )+\sin(5\theta )}{16))}$ ${\displaystyle \cos ^{5}\theta ={\frac {10\cos \theta +5\cos(3\theta )+\cos(5\theta )}{16))}$ ${\displaystyle \sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin(2\theta )-5\sin(6\theta )+\sin(10\theta )}{512))}$

In general terms of powers of ${\displaystyle \sin \theta }$ or ${\displaystyle \cos \theta }$ the following is true, and can be deduced using De Moivre's formula, Euler's formula and the binomial theorem.

if n is ... ${\displaystyle \cos ^{n}\theta }$ ${\displaystyle \sin ^{n}\theta }$
n is odd ${\displaystyle \cos ^{n}\theta ={\frac {2}{2^{n))}\sum _{k=0}^{\frac {n-1}{2)){\binom {n}{k))\cos ((\big (}(n-2k)\theta {\big )))}$ ${\displaystyle \sin ^{n}\theta ={\frac {2}{2^{n))}\sum _{k=0}^{\frac {n-1}{2))(-1)^{\left({\frac {n-1}{2))-k\right)}{\binom {n}{k))\sin ((\big (}(n-2k)\theta {\big )))}$
n is even ${\displaystyle \cos ^{n}\theta ={\frac {1}{2^{n))}{\binom {n}{\frac {n}{2))}+{\frac {2}{2^{n))}\sum _{k=0}^((\frac {n}{2))-1}{\binom {n}{k))\cos ((\big (}(n-2k)\theta {\big )))}$ ${\displaystyle \sin ^{n}\theta ={\frac {1}{2^{n))}{\binom {n}{\frac {n}{2))}+{\frac {2}{2^{n))}\sum _{k=0}^((\frac {n}{2))-1}(-1)^{\left({\frac {n}{2))-k\right)}{\binom {n}{k))\cos ((\big (}(n-2k)\theta {\big )))}$

Product-to-sum and sum-to-product identities

The product-to-sum identities[27] or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations.[28] See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.

Product-to-sum identities

• ${\displaystyle \cos \theta \,\cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi ) \over 2))$
• ${\displaystyle \sin \theta \,\sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi ) \over 2))$
• ${\displaystyle \sin \theta \,\cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi ) \over 2))$
• ${\displaystyle \cos \theta \,\sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi ) \over 2))$
• ${\displaystyle \tan \theta \,\tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )))}$
• ${\displaystyle \tan \theta \,\cot \varphi ={\frac {\sin(\theta +\varphi )+\sin(\theta -\varphi )}{\sin(\theta +\varphi )-\sin(\theta -\varphi )))}$
• {\displaystyle {\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n))}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{where ))e=(e_{1},\ldots ,e_{n})\in S=\{1,-1\}^{n}\end{aligned))}
• ${\displaystyle \prod _{k=1}^{n}\sin \theta _{k}={\frac {(-1)^{\left\lfloor {\frac {n}{2))\right\rfloor )){2^{n))}{\begin{cases}\displaystyle \sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;{\text{if))\;n\;{\text{is even)),\\\displaystyle \sum _{e\in S}\sin(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;{\text{if))\;n\;{\text{is odd))\end{cases))}$

Sum-to-product identities

The sum-to-product identities are as follows:[29]

• ${\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2))\right)\cos \left({\frac {\theta \mp \varphi }{2))\right)}$
• ${\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2))\right)\cos \left({\frac {\theta -\varphi }{2))\right)}$
• ${\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\frac {\theta +\varphi }{2))\right)\sin \left({\frac {\theta -\varphi }{2))\right)}$
• ${\displaystyle \tan \theta \pm \tan \varphi ={\frac {\sin(\theta \pm \varphi )}{\cos \theta \,\cos \varphi ))}$

Hermite's cotangent identity

 Main article: Hermite's cotangent identity

Charles Hermite demonstrated the following identity.[30] Suppose ${\displaystyle a_{1},\ldots ,a_{n))$ are complex numbers, no two of which differ by an integer multiple of π. Let

${\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix))\cot(a_{k}-a_{j})}$

(in particular, ${\displaystyle A_{1,1},}$ being an empty product, is 1). Then

${\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2))+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).}$

The simplest non-trivial example is the case n = 2:

${\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).}$

Finite products of trigonometric functions

For coprime integers n, m

${\displaystyle \prod _{k=1}^{n}\left(2a+2\cos \left({\frac {2\pi km}{n))+x\right)\right)=2\left(T_{n}(a)+{(-1)}^{n+m}\cos(nx)\right)}$

where Tn is the Chebyshev polynomial.[citation needed]

The following relationship holds for the sine function

${\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n))\right)={\frac {n}{2^{n-1))}.}$

More generally for an integer n > 0[31]

${\displaystyle \sin(nx)=2^{n-1}\prod _{k=0}^{n-1}\sin \left({\frac {k}{n))\pi +x\right)=2^{n-1}\prod _{k=1}^{n}\sin \left({\frac {k}{n))\pi -x\right).}$

or written in terms of the chord function ${\textstyle \operatorname {crd} x\equiv 2\sin {\tfrac {1}{2))x}$,

${\displaystyle \operatorname {crd} (nx)=\prod _{k=1}^{n}\operatorname {crd} \left({\frac {k}{n))2\pi -x\right).}$

This comes from the factorization of the polynomial ${\textstyle z^{n}-1}$ into linear factors (cf. root of unity): For a point z on the complex unit circle and an integer n > 0,

${\displaystyle z^{n}-1=\prod _{k=1}^{n}z-\exp {\Bigl (}{\frac {k}{n))2\pi i{\Bigr )}.}$

Linear combinations

For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of ${\displaystyle c}$ and ${\displaystyle \varphi }$.

Sine and cosine

The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,[32][33]

${\displaystyle a\cos x+b\sin x=c\cos(x+\varphi )}$

where ${\displaystyle c}$ and ${\displaystyle \varphi }$ are defined as so:

{\displaystyle {\begin{aligned}c&=\operatorname {sgn}(a){\sqrt {a^{2}+b^{2))},\\\varphi &={\arctan }{\bigl (}{-b/a}{\bigr )},\end{aligned))}

given that ${\displaystyle a\neq 0.}$

Arbitrary phase shift

More generally, for arbitrary phase shifts, we have

${\displaystyle a\sin(x+\theta _{a})+b\sin(x+\theta _{b})=c\sin(x+\varphi )}$

where ${\displaystyle c}$ and ${\displaystyle \varphi }$ satisfy:

{\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}+2ab\cos \left(\theta _{a}-\theta _{b}\right),\\\tan \varphi &={\frac {a\sin \theta _{a}+b\sin \theta _{b)){a\cos \theta _{a}+b\cos \theta _{b))}.\end{aligned))}

More than two sinusoids

The general case reads[33]

${\displaystyle \sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta ),}$
where
${\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}$
and
${\displaystyle \tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i)){\sum _{i}a_{i}\cos \theta _{i))}.}$

Lagrange's trigonometric identities

These identities, named after Joseph Louis Lagrange, are:[34][35][36]

{\displaystyle {\begin{aligned}\sum _{k=0}^{n}\sin k\theta &={\frac {\cos {\tfrac {1}{2))\theta -\cos \left(\left(n+{\tfrac {1}{2))\right)\theta \right)}{2\sin {\tfrac {1}{2))\theta ))\\[5pt]\sum _{k=0}^{n}\cos k\theta &={\frac {\sin {\tfrac {1}{2))\theta +\sin \left(\left(n+{\tfrac {1}{2))\right)\theta \right)}{2\sin {\tfrac {1}{2))\theta ))\end{aligned))}
for ${\displaystyle \theta \not \equiv 0{\pmod {2\pi )).}$

A related function is the Dirichlet kernel:

${\displaystyle D_{n}(\theta )=1+2\sum _{k=1}^{n}\cos k\theta ={\frac {\sin \left(\left(n+{\tfrac {1}{2))\right)\theta \right)}{\sin {\tfrac {1}{2))\theta )).}$

A similar identity is[37]

${\displaystyle \sum _{k=1}^{n}\cos(2k-1)\alpha ={\frac {\sin(2n\alpha )}{2\sin \alpha )).}$

The proof is the following. By using the angle sum and difference identities,

${\displaystyle \sin(A+B)-\sin(A-B)=2\cos A\sin B.}$
Then let's examine the following formula,

${\displaystyle 2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha =2\sin \alpha \cos \alpha +2\sin \alpha \cos 3\alpha +2\sin \alpha \cos 5\alpha +\ldots +2\sin \alpha \cos(2n-1)\alpha }$
and this formula can be written by using the above identity,

{\displaystyle {\begin{aligned}&2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha \\&\quad =\sum _{k=1}^{n}(\sin(2k\alpha )-\sin(2(k-1)\alpha ))\\&\quad =(\sin 2\alpha -\sin 0)+(\sin 4\alpha -\sin 2\alpha )+(\sin 6\alpha -\sin 4\alpha )+\ldots +(\sin(2n\alpha )-\sin(2(n-1)\alpha ))\\&\quad =\sin(2n\alpha ).\end{aligned))}

So, dividing this formula with ${\displaystyle 2\sin \alpha }$ completes the proof.

Certain linear fractional transformations

If ${\displaystyle f(x)}$ is given by the linear fractional transformation

${\displaystyle f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha )),}$
and similarly
${\displaystyle g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta )),}$
then
${\displaystyle f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac ((\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}((\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta ))).}$

More tersely stated, if for all ${\displaystyle \alpha }$ we let ${\displaystyle f_{\alpha ))$ be what we called ${\displaystyle f}$ above, then

${\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.}$

If ${\displaystyle x}$ is the slope of a line, then ${\displaystyle f(x)}$ is the slope of its rotation through an angle of ${\displaystyle -\alpha .}$

Relation to the complex exponential function

 Main article: Euler's formula

Euler's formula states that, for any real number x:[38]

${\displaystyle e^{ix}=\cos x+i\sin x,}$
where i is the imaginary unit. Substituting −x for x gives us:
${\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x.}$

These two equations can be used to solve for cosine and sine in terms of the exponential function. Specifically,[39][40]

${\displaystyle \cos x={\frac {e^{ix}+e^{-ix)){2))}$
${\displaystyle \sin x={\frac {e^{ix}-e^{-ix)){2i))}$

These formulae are useful for proving many other trigonometric identities. For example, that ei(θ+φ) = e e means that

cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).

That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.

The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the complex logarithm.

Function Inverse function[41]
${\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta )){2i))}$ ${\displaystyle \arcsin x=-i\,\ln \left(ix+{\sqrt {1-x^{2))}\right)}$
${\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta )){2))}$ ${\displaystyle \arccos x=-i\ln \left(x+{\sqrt {x^{2}-1))\right)}$
${\displaystyle \tan \theta =-i\,{\frac {e^{i\theta }-e^{-i\theta )){e^{i\theta }+e^{-i\theta ))))$ ${\displaystyle \arctan x={\frac {i}{2))\ln \left({\frac {i+x}{i-x))\right)}$
${\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta ))))$ ${\displaystyle \operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x))+{\sqrt {1-{\frac {1}{x^{2))))}\right)}$
${\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta ))))$ ${\displaystyle \operatorname {arcsec} x=-i\,\ln \left({\frac {1}{x))+i{\sqrt {1-{\frac {1}{x^{2))))}\right)}$
${\displaystyle \cot \theta =i\,{\frac {e^{i\theta }+e^{-i\theta )){e^{i\theta }-e^{-i\theta ))))$ ${\displaystyle \operatorname {arccot} x={\frac {i}{2))\ln \left({\frac {x-i}{x+i))\right)}$
${\displaystyle \operatorname {cis} \theta =e^{i\theta ))$ ${\displaystyle \operatorname {arccis} x=-i\ln x}$

Series expansion

When using a power series expansion to define trigonometric functions, the following identities are obtained:[42]

${\displaystyle \sin x=x-{\frac {x^{3)){3!))+{\frac {x^{5)){5!))-{\frac {x^{7)){7!))\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1)){(2n+1)!)),}$
${\displaystyle \cos x=1-{\frac {x^{2)){2!))+{\frac {x^{4)){4!))-{\frac {x^{6)){6!))+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n)){(2n)!)).}$

Infinite product formulae

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:[43][44]

{\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2)){\pi ^{2}n^{2))}\right),&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2)){\pi ^{2}\left(n-{\frac {1}{2))\right)\!{\vphantom {)))^{2))}\right),\\[10mu]\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2)){\pi ^{2}n^{2))}\right),&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2)){\pi ^{2}\left(n-{\frac {1}{2))\right)\!{\vphantom {)))^{2))}\right).\end{aligned))}

Inverse trigonometric functions

 Main article: Inverse trigonometric functions

The following identities give the result of composing a trigonometric function with an inverse trigonometric function.[45]

{\displaystyle {\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2))}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2))))\\\sin(\arccos x)&={\sqrt {1-x^{2))}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2))}{x))\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2))))&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2))))&\tan(\arctan x)&=x\\\sin(\operatorname {arccsc} x)&={\frac {1}{x))&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1)){x))&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1))}\\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1)){x))&\cos(\operatorname {arcsec} x)&={\frac {1}{x))&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1))\\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2))))&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2))))&\tan(\operatorname {arccot} x)&={\frac {1}{x))\\\end{aligned))}

Taking the multiplicative inverse of both sides of the each equation above results in the equations for ${\displaystyle \csc ={\frac {1}{\sin )),\;\sec ={\frac {1}{\cos )),{\text{ and ))\cot ={\frac {1}{\tan )).}$ The right hand side of the formula above will always be flipped. For example, the equation for ${\displaystyle \cot(\arcsin x)}$ is:

${\displaystyle \cot(\arcsin x)={\frac {1}{\tan(\arcsin x)))={\frac {1}{\frac {x}{\sqrt {1-x^{2))))}={\frac {\sqrt {1-x^{2))}{x))}$
while the equations for ${\displaystyle \csc(\arccos x)}$ and ${\displaystyle \sec(\arccos x)}$ are:
${\displaystyle \csc(\arccos x)={\frac {1}{\sin(\arccos x)))={\frac {1}{\sqrt {1-x^{2))))\qquad {\text{ and ))\quad \sec(\arccos x)={\frac {1}{\cos(\arccos x)))={\frac {1}{x)).}$

The following identities are implied by the reflection identities. They hold whenever ${\displaystyle x,r,s,-x,-r,{\text{ and ))-s}$ are in the domains of the relevant functions.

{\displaystyle {\begin{alignedat}{9}{\frac {\pi }{2))~&=~\arcsin(x)&&+\arccos(x)~&&=~\arctan(r)&&+\operatorname {arccot}(r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arccsc}(s)\\[0.4ex]\pi ~&=~\arccos(x)&&+\arccos(-x)~&&=~\operatorname {arccot}(r)&&+\operatorname {arccot}(-r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arcsec}(-s)\\[0.4ex]0~&=~\arcsin(x)&&+\arcsin(-x)~&&=~\arctan(r)&&+\arctan(-r)~&&=~\operatorname {arccsc}(s)&&+\operatorname {arccsc}(-s)\\[1.0ex]\end{alignedat))}

Also,[46]

{\displaystyle {\begin{aligned}\arctan x+\arctan {\dfrac {1}{x))&={\begin{cases}{\frac {\pi }{2)),&{\text{if ))x>0\\-{\frac {\pi }{2)),&{\text{if ))x<0\end{cases))\\\operatorname {arccot} x+\operatorname {arccot} {\dfrac {1}{x))&={\begin{cases}{\frac {\pi }{2)),&{\text{if ))x>0\\{\frac {3\pi }{2)),&{\text{if ))x<0\end{cases))\\\end{aligned))}
${\displaystyle \arccos {\frac {1}{x))=\operatorname {arcsec} x\qquad {\text{ and ))\qquad \operatorname {arcsec} {\frac {1}{x))=\arccos x}$
${\displaystyle \arcsin {\frac {1}{x))=\operatorname {arccsc} x\qquad {\text{ and ))\qquad \operatorname {arccsc} {\frac {1}{x))=\arcsin x}$

The arctangent function can be expanded as a series:[47]

${\displaystyle \arctan(nx)=\sum _{m=1}^{n}\arctan {\frac {x}{1+(m-1)mx^{2))))$

Identities without variables

In terms of the arctangent function we have[46]

${\displaystyle \arctan {\frac {1}{2))=\arctan {\frac {1}{3))+\arctan {\frac {1}{7)).}$

The curious identity known as Morrie's law,

${\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8)),}$

is a special case of an identity that contains one variable:

${\displaystyle \prod _{j=0}^{k-1}\cos \left(2^{j}x\right)={\frac {\sin \left(2^{k}x\right)}{2^{k}\sin x)).}$

Similarly,

${\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3)){8))}$
is a special case of an identity with ${\displaystyle x=20^{\circ ))$:
${\displaystyle \sin x\cdot \sin \left(60^{\circ }-x\right)\cdot \sin \left(60^{\circ }+x\right)={\frac {\sin 3x}{4)).}$

For the case ${\displaystyle x=15^{\circ ))$,

{\displaystyle {\begin{aligned}\sin 15^{\circ }\cdot \sin 45^{\circ }\cdot \sin 75^{\circ }&={\frac {\sqrt {2)){8)),\\\sin 15^{\circ }\cdot \sin 75^{\circ }&={\frac {1}{4)).\end{aligned))}

For the case ${\displaystyle x=10^{\circ ))$,

${\displaystyle \sin 10^{\circ }\cdot \sin 50^{\circ }\cdot \sin 70^{\circ }={\frac {1}{8)).}$

The same cosine identity is

${\displaystyle \cos x\cdot \cos \left(60^{\circ }-x\right)\cdot \cos \left(60^{\circ }+x\right)={\frac {\cos 3x}{4)).}$

Similarly,

{\displaystyle {\begin{aligned}\cos 10^{\circ }\cdot \cos 50^{\circ }\cdot \cos 70^{\circ }&={\frac {\sqrt {3)){8)),\\\cos 15^{\circ }\cdot \cos 45^{\circ }\cdot \cos 75^{\circ }&={\frac {\sqrt {2)){8)),\\\cos 15^{\circ }\cdot \cos 75^{\circ }&={\frac {1}{4)).\end{aligned))}

Similarly,

{\displaystyle {\begin{aligned}\tan 50^{\circ }\cdot \tan 60^{\circ }\cdot \tan 70^{\circ }&=\tan 80^{\circ },\\\tan 40^{\circ }\cdot \tan 30^{\circ }\cdot \tan 20^{\circ }&=\tan 10^{\circ }.\end{aligned))}

The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):

${\displaystyle \cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2)).}$

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:

${\displaystyle \cos {\frac {2\pi }{21))+\cos \left(2\cdot {\frac {2\pi }{21))\right)+\cos \left(4\cdot {\frac {2\pi }{21))\right)+\cos \left(5\cdot {\frac {2\pi }{21))\right)+\cos \left(8\cdot {\frac {2\pi }{21))\right)+\cos \left(10\cdot {\frac {2\pi }{21))\right)={\frac {1}{2)).}$

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.

Other cosine identities include:[48]

{\displaystyle {\begin{aligned}2\cos {\frac {\pi }{3))&=1,\\2\cos {\frac {\pi }{5))\times 2\cos {\frac {2\pi }{5))&=1,\\2\cos {\frac {\pi }{7))\times 2\cos {\frac {2\pi }{7))\times 2\cos {\frac {3\pi }{7))&=1,\end{aligned))}
and so forth for all odd numbers, and hence
${\displaystyle \cos {\frac {\pi }{3))+\cos {\frac {\pi }{5))\times \cos {\frac {2\pi }{5))+\cos {\frac {\pi }{7))\times \cos {\frac {2\pi }{7))\times \cos {\frac {3\pi }{7))+\dots =1.}$

Many of those curious identities stem from more general facts like the following:[49]

${\displaystyle \prod _{k=1}^{n-1}\sin {\frac {k\pi }{n))={\frac {n}{2^{n-1))))$
and
${\displaystyle \prod _{k=1}^{n-1}\cos {\frac {k\pi }{n))={\frac {\sin {\frac {\pi n}{2))}{2^{n-1))}.}$

Combining these gives us

${\displaystyle \prod _{k=1}^{n-1}\tan {\frac {k\pi }{n))={\frac {n}{\sin {\frac {\pi n}{2))))}$

If n is an odd number (${\displaystyle n=2m+1}$) we can make use of the symmetries to get

${\displaystyle \prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1))={\sqrt {2m+1))}$

The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved:

${\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n))=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n))={\frac {\sqrt {2)){2^{n))))$

Computing π

An efficient way to compute π to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula:

${\displaystyle {\frac {\pi }{4))=4\arctan {\frac {1}{5))-\arctan {\frac {1}{239))}$