# 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). Remember!

• 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. Complementary Angle Identities

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

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