Mathematical identities are equations that hold true for all values of the variables involved. They express a relationship between different mathematical expressions and are often used to simplify calculations or prove other mathematical statements. Common examples include the Pythagorean theorem and trigonometric identities. These identities are essential tools in various fields of mathematics, including algebra, geometry, and calculus. They help mathematicians and students manipulate equations and solve problems more efficiently. Understanding and applying these identities can lead to deeper insights into mathematical concepts and their applications in real-world scenarios.