Verifying Identities

An identity is defined as an equality that holds true regardless of its variables. This means that no matter what value is used for any variable in the equation, both sides of the equation will always be equal to each other.  For example, consider the equation. The equation can be simplified as follows:

Therefore, any value will satisfy the equation, making it an identity.

Trigonometric Identities
There are eleven fundamental trigonometric identities grouped into three categories: reciprocal identities, quotient identities, and pythagorean identities.

Reciprocal Identities:
    






Quotient Identities:

Pythagorean Identities:






These fundamental identities can be used to verify other trigonometric identities by substitution. Verifying identities means to transform one side of the equation into the other
Example:

Because, . We can use this substitution to make both sides of the equation look identical, thereby verifying the identity.

All trigonometric identities can be verified by using the eleven fundamental identities, or by using algebra. Here's a more complicated problem: