# 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.

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.

Remember!

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

Solution:

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

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