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!
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.
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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