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The difference between identities and formulas in mathematics is as follows:
Identities:
An identity is a mathematical statement that is true for all values of the variables involved.
Identities express a fundamental relationship between mathematical expressions that holds true regardless of the specific values substituted.
Identities are denoted using the symbol “≡” (triple bar) to indicate that the equality is true for all possible values.
Examples of identities:
(a + b)² ≡ a² + 2ab + b²
sin²(x) + cos²(x) ≡ 1
d/dx(x^n) ≡ nx^(n1)
Formulas:
A formula is a mathematical expression that describes a specific relationship between variables or quantities.
Formulas are used to calculate or compute a specific value or result based on the given inputs or variables.
Formulas are typically denoted using the “=” symbol to indicate the equality between the expression on the lefthand side and the value or expression on the righthand side.
Formulas are often contextdependent and may only hold true under certain conditions or assumptions.
Examples of formulas:
Area of a rectangle = length × width
Pythagorean Theorem: a² + b² = c²
Force = mass × acceleration
Volume of a sphere = (4/3) × π × r³
The key differences are:
Identities are true for all values of the variables, while formulas are specific relationships that may only hold true under certain conditions.
Identities are denoted using the “≡” symbol, while formulas use the “=” symbol.
Identities express fundamental mathematical relationships, while formulas are used to calculate or compute specific values or results.
In summary, identities are universal mathematical statements, while formulas are contextdependent expressions used for calculation or computation.