Algebraic Identity
In algebra, an equality that is true regardless of the numerical values assigned to its variables.
Symbol
The symbol "≡" is sometimes used for trigonometric identities.Properties
These are special algebraic identities :- [latex]{(a+b)^2}[/latex] = [latex]{a^2}[/latex]+ 2ab + [latex]{b^2}[/latex]
- [latex]{(a–b)^2}[/latex] = [latex]{a^2}[/latex]– 2ab + [latex]{b^2}[/latex]
- [latex]{(a–b)^3}[/latex] = [latex]{a^3}[/latex]– 3a[latex]{b^2}[/latex]+ 3[latex]{a^2}[/latex]b – [latex]{b^3}[/latex]
- [latex]{(a+b)^3}[/latex] = [latex]{a^3}[/latex]+ 3a[latex]{b^2}[/latex]+ 3[latex]{a^2}[/latex]b + [latex]{b^3}[/latex]
- [latex]{a^2}[/latex] – [latex]{b^2}[/latex] = (a + b) (a – b)
- [latex]{a^3}[/latex] – [latex]{b^3}[/latex] = (a – b)([latex]{a^2}[/latex] + ab + [latex]{b^2}[/latex])
- [latex]{a^3}[/latex] + [latex]{b^3}[/latex] = (a + b)([latex]{a^2}[/latex] – ab + [latex]{b^2}[/latex])
Examples
- The equality (a + b)² = a² + 2ab + b² is an algebraic identity.
- The relation sin (A + B) ≡ sin A cos B + sin B cos A is a trigonometric identity.