Trigonometric Identities

The trigonometric identities are the equations which are valid for the right-angled triangles. The trigonometric ratios are the basic identities in trigonometry which are applied throughout the concept.

The fundamental identities of trigonometry are as follows:

To obtain additional trigonometric identities, we use Pythagoras theorem.

Pythagoras Theorem statement:

“In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.”

From the image above, we have
c²=a²+b² (1)

Divide throughout equation (1) by c².

Let us write trigonometric ratios for the image given above. Assume angle to be θ.

Substitute sine and cosine values in equation (2).


  • implies that, first find the sine of θ and then square its result.
  • sin 2 implies that, first find the square of θ and then find the sine of the result.
  • Using the result obtained from Pythagoras theorem + =1, we get the following identities.

    Negative Angle Identities

    Complementary Angle Identities

    Each of the six trigonometric functions is the same as its complementary angle co-function.

    Double Angle Identities

    Half Angle Identities

    When angle θ is reduced to half, we get θ/2. Identities for half angles are as follows.


    Angle Sum, Difference, and Product Identities

    All the above identities holds good for only right-angled triangles. There are certain identities that hold for all triangles. The identities are as follows.

  • Sine rule or Law of Sines
  • Cosine rule or Law of Cosines
  • Law of Tangents
    Consider the triangle shown below.

    Law of Sines:

    Law of Cosines:

    The above equation can be rearranged as cos C =

    Law of Tangents:

    The identities are applied to prove many equations. Let us learn some of the examples and understand the applications of these identities learnt so far.

    Example 1: Prove that


    Example 2: Prove that

    Example 3: Find the value of the following.

    Solution: 3θ can be written as 2θ + θ.

    Know more about other concepts of Trigonometry on What is Trigonometry – An Introduction and Trigonometric Ratios

    Online Math Classes > Math Concepts > Trigonometry > Trigonometric Identities