Identity (mathematics)

In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B produce the same value for all values of the variables within a certain range or domain. In simpler terms, an identity is an equation that is always true, regardless of the specific values you substitute for any variables involved.

Identities are fundamental tools in various branches of mathematics, including algebra, trigonometry, calculus, and discrete mathematics. They provide a way to simplify expressions, solve equations, and prove theorems.

Properties of Identities

• Universally True (within a domain): Unlike regular equations that may only hold for certain specific values of the variables, an identity is true for all permissible values of the variables involved. The permissible values define the domain for which the identity holds.

• Simplification: Identities are often used to simplify complex expressions. By substituting one side of the identity for the other, you can often rewrite an expression in a more manageable form.

• Equation Solving: Identities can be instrumental in solving equations. By using an identity to transform an equation, you can often reduce it to a simpler form that is easier to solve.

• Proofs: Identities are frequently used in mathematical proofs. By manipulating expressions using identities, you can often establish relationships between different mathematical objects.

Types of Identities

Identities exist across many areas of mathematics. Some common examples include:

• Algebraic Identities: Examples include (a+b)^2 = a^2 + 2ab + b^2 and a^2 - b^2 = (a+b)(a-b). These are fundamental to manipulating algebraic expressions.

• Trigonometric Identities: Examples include sin^2(x) + cos^2(x) = 1 and tan(x) = sin(x)/cos(x). These are crucial for simplifying trigonometric expressions and solving trigonometric equations.

• Logarithmic Identities: Examples include log(ab) = log(a) + log(b) and log(a^b) = b*log(a). These are useful for manipulating logarithmic expressions.

• Calculus Identities: Examples include the product rule (d/dx)(uv) = u(dv/dx) + v(du/dx) and integration by parts. These are essential for differentiation and integration.

Distinction from Equations

It's important to differentiate between identities and equations. An equation is a statement of equality that may or may not be true depending on the values of the variables. An identity, however, is always true for all valid values of the variables within its defined domain.