A mathematical identity is an equation that holds true for all values of the variables involved. Unlike an equation that may only be true for specific values, an identity is universally valid. For example, the identity a + b = b + a demonstrates that the order of addition does not affect the sum, showcasing the commutative property. Identities are essential in various branches of mathematics, including algebra and calculus. They help simplify expressions and solve equations. Common examples include the Pythagorean identity in trigonometry, which relates the sides of a right triangle, and the binomial theorem, which expands expressions like (a + b) ^n.