How Do You Prove an Algebraic Identity?

To prove an algebraic identity, begin with one side of the statement and use valid algebraic transformations until it becomes the other side. In IB Mathematics: Analysis and Approaches, the preferred deductive proof structure is usually to transform the more complicated side without assuming the identity is already true.

An identity is an equation that is true for every value in its domain. It is written using the identity symbol $\equiv$, rather than merely $=$.

LHS-to-RHS Method

1. Start with the left-hand side (LHS), or whichever side is more complicated.
2. Apply one valid operation at a time, such as expanding, factorizing, combining fractions, or using an established identity.
3. Continue until the expression is exactly the right-hand side (RHS).
4. Finish by stating that the required identity has been established.

For example, prove that

$$\frac{1}{x+1}+\frac{1}{x(x+1)}\equiv\frac{1}{x},$$

where $x\neq 0,-1$.

Starting with the LHS,

$$\begin{aligned} \frac{1}{x+1}+\frac{1}{x(x+1)} &\equiv \frac{x}{x(x+1)}+\frac{1}{x(x+1)}\\ &\equiv \frac{x+1}{x(x+1)}\\ &\equiv \frac{1}{x}. \end{aligned}$$

Therefore, LHS $\equiv$ RHS, so the identity is proved for all values in its domain.

The domain restriction matters because cancelling $x+1$ is valid only when $x+1\neq 0$. More generally, each line must preserve the expression on its stated domain. Avoid dividing by an expression that could equal zero, taking an unjustified square root, or applying logarithm laws when their arguments are not positive. These steps may change the solution set rather than prove equivalence.

A common misconception is to begin with the complete statement and manipulate both sides simultaneously. This assumes what must be proved and can create circular reasoning. Instead, keep a clear one-direction chain from one side to the other.

Exam technique: This is SL 1.6, Simple deductive proof, and is suited to analytic Paper 1 work. For a command such as show that, examiners expect every important algebraic step, correct use of $\equiv$, and any necessary domain restrictions.