Trigonometric Identities

A function is a mathematical statement relating the different variables. When we assign the values for a set of independent variables, we get the definite value for the dependent variable.
Identity is a mathematical expression, which is valid for all the values of the variables. When the identity has terms involving the algebraic expression we call it as algebraic identity and the function involving the trigonometric expressions are called trigonometric identities.
Verifying Trigonometric identities:
Identities are mathematical statements that are valid for all the values of the variables. For example
(a +b)^2 = a^2 +2ab +b^2

The above expression is valid for all the values of ‘a’ and ‘b’. Trigonometry is the field of study involving triangles’ sides and angles. If the identity has any of the trigonometric functions, then it is called trigonometric identity. As the trigonometric functions are related to the angles, we substitute the angles in the functions of identities. We adopt the following steps to verify an identity.
Step 1: Choose an angle from the defined domain.
Step 2: Replace the variable by the chosen value
Step 3: Evaluate the function and simplify
Example: Sin^2 (x) + cos 2(x) =1
Step 1: For the value of x = 90
Step 2: Sin^2 (90) + cos^2 (90)
Step 3: 1+ 0 =1. Hence verified

Fundamental Trigonometric Identities
Trigonometry, a field of study involving the sides and angles of a triangle has a set of fundamental definition of trigonometric functions. Using the theorem of Pythagoras, we have three fundamental trigonometric identities called as Pythagorean identities, which are as follows.

In a triangle ABC:
(1) 1 = sin^2 (A) +cos^2 (A)
(2) 1+ tan^2(A) = sec^2(A)
(3) 1+ cot^2(A) = cosec^2(A)

Simplifying Trigonometric Identities
To simplify a given trigonometric identity, we always rely on the algebraic methods. Especially, we adopt PEMDAS/ BOADMAS in simplifying the identities.  In addition to this, we use the above three identities in simplifying the given trigonometric identities. At the same time, we ensure that the defined trigonometric identity is a valid one in the defined domain of variables.

Table of Trigonometric Identities:
The fundamental trigonometric identities are as follows
Reciprocal Identities:
Sec(A) = 1/cos(A) Cosec (A) = 1/ Sin(A) tan(A) = sin(A)/cos(A),   cot(A) = 1/ Tan(A)= cos(A)/sin(A)

Pythagorean Identities
(1) 1 = sin^2 (A) +cos^2 (A) (2) 1+ tan^2(A) = sec^2(A) (3) 1+ cot^2(A) = cosec^2(A)

Even-Odd Identities
(1) Sin (-A) = -sin(A) (2) cos(-A) = cos (A) (3) tan(-A) = -tan (A)
(4) Cosec (-A)= -cosec(A) (5) sec(-A)= -sec A (6) cot(-A) = -cot(A)