23. Equations vs. Identities

In mathematics, we encounter terms, equations, expressions, inequalities and identities.

 

• $7xy$ is a term.
• $5x-2=8$ is an equation.
• $7x^2+3x$ is an expression.
• $5x-2<8$ is an inequality.
• $2x(x-4)\equiv 2x^2-8x$ is an identity.

 

In particular, the difference between equations and identities is often subtle and confusing. In short, we use an equality sign when the statement is true for some values of the variable, say $x$, while we use an identity sign when the statement holds true for all values.

 

For example, $$ \begin{align} 2x-1=0 \end{align} $$ is an equation as it is true only for $x=\frac12$. On the other hand, $$ \begin{align} \cos^2x+\sin^2x\equiv 1 \end{align} $$ is an identity as it is true for all values of $x$.

 

Example. Some other identities are:

$$ \begin{align} {\rm (1)}&\qquad 4r^3\equiv r^2(r+1)^2-r^2(r-1)^2 \\ {\rm (2)}&\qquad\frac{1}{r(r+1)}\equiv\frac1{r}-\frac1{r+1} \end{align} $$