The trigonometric identites are a set of equations derived from the properties of the circle and the trigonometric functions. The equations are useful for manipulating and transforming math expressions.
Name  Identity 

Tangent Identity  
Cotangent Identity  
Pythagorean’s Identity  
Sum of Two Angles (Sine)  
Sum of Two Angles (Cosine)  
Difference of Two Angles (Sine)  
Difference of Two Angles (Cosine)  
DoubleAngle Identity (Sine)  
DoubleAngle Identity (Cosine)  
DoubleAngle Identity (Cosine 2)  
DoubleAngle Identity (Cosine 3)  
HalfAngle Identity (Cosine)  
HalfAngle Identity (Sine) 
Below, each identity is summarized and linked to examples that show how to derive the identity. In general, there are three strategies for deriving the identities:
 Use purely trigonometry and algebra.^{[1]}
 Use the geometry of the circle.^{[2]}
 Use the geometry of the complex plane^{[3]}.
It is helpful to be familiar with the definitions of the trigonometric functions, the equation for Pythagorean Theorem and the trigonometry of the unit circle.
The tangent identity shows how the tangent function can be defined by the cosine and sine functions. The simplest way to see this connection is to define the length of the adjacent and opposite sides of the right triangle in terms of the and of the angle and the hypotenuse of the triangle denoted with the variable .
Then, by substituting these lengths into the definition of tangent, the length of the hypotenuse cancels, leaving the tangent identity.
Geometrically, the tangent length of an angle is equal to the length of the line tangent to the point the angle describes on the unit circle and the point where the tangent line intersects the xaxis. This length can be solved for using sine and cosine to derive the same formula.
The graph of tangent illustrates how the tangent length changes with respect to the angle . Note, the graph of tangent diverges to infinity where the cosine function is .
The cotangent identity expresses how the cotangent of an angle is the reciprocal of the tangent of an angle. On the unit circle, the cotangent of an angle is equal to the adjacent side of a right triangle with an opposite side of length . Geometrically, the cotangent the corresponding length on the tangent line to the point on the circle.
The graph of cotangent and tangent shows the reciprocal relationship between the two functions. As tangent approaches so does cotangent and as tangent approaches cotangent divereges to infinity.
The pythagorean identity relates Pythagorean’s theorem to the geometry of a righttriangle on the unit circle. Pythagorean’s theorem equates the squared lengths of a right triangle together:
The cosine and sine of an angle give the coordinates of a point along the unit circle. Substituting these lengths into the equation for pythagorean’s theorem yields Pythagorean’s identity.
The sum of two angles identities express the cosine and sine of the sum in terms of the sine and cosine of the individual angles. The identities can be derived three ways: 1) By using the geometry of a right triangle. 2) By using the geometry of the righttriangle on the circle. 3) By using the geometry of the complex plane.
The figure above demonstrates how the identities can be formulated purely from the geometry of the righttriangle. See all derivations below:
To derive the sum of two angles identities, two right triangles are placed next to eachother so their angles sum together, then their proportions are related together.
The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.
The sum of two angles identities can be derived using the properties of the complex plane and Euler's formula.
The difference of two angles identities express the cosine and sine of the difference in terms of the sine and cosine of the individual angles. The “proof without words” below demonstrates one way to derive the identities.
To derive the idientities stepbystep using the geometry of the image above see the example below:
To derive the difference of two angles identities, two right triangles are placed next to eachother so their angles sum together to be one angle and one triangle's angle is the difference of the sum and the other.
Derive the difference of two angle identities using the properties of complex numbers and Euler's formula in the complex plane.
The double angle identities express the cosine and sine of a double angle in terms of the sine and cosine of the single angle. The identities can be derived three ways: 1) By using the previously derived theorems on this page such as Pythagorean’s Identity and the Sum of Two Angles identities. 2) By using the geometry of the inscribed angle theorem and the formula for area of a triangle. 3) By using the complex plane and the properties of complex numbers.
The figure above demonstrates the inscribed angle theorem and the properties of similar triangles can be used to derive the double angle identities:
This example derives the double angle identities using algebra and the sum of two angles identities.
The double angle identities can be derived using the inscribed angle theorem on the circle of radius one.
The trigonometric double angle identites can be derived using the properties of the complex plane.
The half angle identities express the cosine and sine of a halfangle in terms of the sine and cosine of the single angle. The identities can be derived using the geometry of the inscribed angle shown below:
The halfangle identities can also be derived using the double angle formulas and the previous theorems derived above.
The halfangle identities can be derived from the double angle identities by transforming the angles using algebra and then solving for the halfangle expression.
This example demosntrates how to derive the trig. half angle identities using the inscribed angle theorem.
This example derives the trig. half angle identities in the complex plane.
There are six trigonometric functions that relate to the geometry of the righttriangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.
The tangent function can be expressed as the ratio of sine over cosine of an angle.
The cotangent function can be expressed as the reciprocal of the tangent function.
The pythagorean identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is derived using pythagorean's theorem and the properties of the unit circle.
The sum of two angles identities express the cosine and sine of the sum of two angles in terms of their individual cosine and sine components.
The difference of two angles identities express the cosine and sine of the difference of the two angles in terms of their individual components.
The double angle identities give the sine and cosine of a double angle in terms of the sine and cosine of a single angle.
The half angle identities give the sine and cosine of a half angle in terms of the sine and cosine of an angle.
The pythagorean theorem equates the square of the sides of a right triangle together.
The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.

Derive the Trigonometric IdentitiesWumbo (internal)

Derive the Trigonometric Identities (Unit Circle)Wumbo (internal)

Derive the Trigonometric Identities (Complex Plane)Wumbo (internal)