List of trigonometric identities

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

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

sin

2

⁡

θ

+

cos

2

⁡

θ

=

1

,

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

where

sin

2

⁡

θ

{\displaystyle \sin ^{2}\theta }

means

(

sin

⁡

θ

)

2

{\displaystyle {(\sin \theta )}^{2}}

and

cos

2

⁡

θ

{\displaystyle \cos ^{2}\theta }

means

(

cos

⁡

θ

)

2

.

{\displaystyle {(\cos \theta )}^{2}.}

This can be viewed as a version of the Pythagorean theorem, and follows from the equation

x

2

+

y

2

=

1

{\displaystyle x^{2}+y^{2}=1}

for the unit circle. This equation can be solved for either the sine or the cosine:

sin

⁡

θ

=

±

1

−

cos

2

⁡

θ

,

cos

⁡

θ

=

±

1

−

sin

2

⁡

θ

.

{\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

θ

.

{\displaystyle \theta .}

Dividing this identity by

sin

2

⁡

θ

{\displaystyle \sin ^{2}\theta }

,

cos

2

⁡

θ

{\displaystyle \cos ^{2}\theta }

, or both yields the following identities:

1

+

cot

2

⁡

θ

=

csc

2

⁡

θ

1

+

tan

2

⁡

θ

=

sec

2

⁡

θ

sec

2

⁡

θ

+

csc

2

⁡

θ

=

sec

2

⁡

θ

csc

2

⁡

θ

{\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):

Reflections, shifts, and periodicity

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