Trigonometric identities are fundamental concepts in mathematics, particularly in trigonometry. These identities are equations involving trigonometric functions that are true for all values of the variable where the functions are defined. Understanding these identities is crucial for solving various mathematical problems and for applications in fields such as physics, engineering, and computer science.

### Basic Trigonometric Identities

The most basic trigonometric identities include the Pythagorean identities, such as sin²(θ) + cos²(θ) = 1. These identities are derived from the Pythagorean theorem and are essential in simplifying trigonometric expressions and solving equations.

### Angle Sum and Difference Identities

Angle sum and difference identities, such as sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β), allow for the simplification of expressions involving the sum or difference of angles. These identities are particularly useful in solving problems involving trigonometric functions of angles that are not immediately apparent.

### Double Angle and Half Angle Identities

Double angle and half angle identities, such as sin(2θ) = 2sin(θ)cos(θ), provide ways to express trigonometric functions of multiple angles in terms of functions of a single angle. These identities help in solving more complex trigonometric equations and are valuable in integration and differentiation.

In summary, trigonometric identities are vital tools in mathematics, enabling the simplification and solution of various problems. Mastery of these identities enhances one’s ability to tackle complex trigonometric functions and their applications effectively.