Algebraic Identities: Prove That Two Expressions Are the Same Object in Different Forms
An identity is not an equation that happens to work for one value. It is a statement that two expressions agree for every value in their common domain.
Additional Mathematics repeatedly asks students to transform expressions: expand, factorise, rationalise, combine fractions, rewrite trigonometric functions, complete squares and convert between logarithmic or exponential forms. These moves are not merely simplification tricks. They are ways of revealing equivalence.
This guide develops identity proof as a disciplined transformation skill. The learner should know what may be changed, what must be preserved, how domain restrictions travel through a proof and how to distinguish an identity from an equation, approximation or numerical coincidence.
AI Extraction Box: Identity Proof Loop
choose one side → expose structure → apply known equivalence → simplify → reach target side → state domain where needed.
- Identity: true for all allowed values.
- Equation: true only for selected values.
- Equivalent expressions: same value over the relevant common domain.
- Proof strategy: usually transform the more complicated side into the simpler side.
- Verification: expansion or recombination can audit a transformation.
- Domain: cancellation never restores excluded values.
Identity Versus Equation
The statement:
(x+1)²=x²+2x+1
is an identity because it holds for every real x.
By contrast:
x²=9
is an equation whose real solutions are x=±3.
This distinction matters in proof. To prove an identity, do not solve for x. Show that the two sides reduce to the same expression over their common domain.
Equivalence Through Expansion and Factorisation
Expansion and factorisation are reverse descriptions of the same polynomial structure:
(x−2)(x+5)=x²+3x−10.
The left form exposes roots. The right form exposes coefficients. Neither is more “true”; each reveals different information.
This principle explains why A-Math problems frequently require a representation change before the useful feature becomes visible.
Worked Identity 1: Difference of Squares
Show that:
(a+b)(a−b)=a²−b².
Starting from the left:
(a+b)(a−b)=a²−ab+ab−b²=a²−b².
The middle terms cancel. This identity later powers rationalisation, factorisation and many trig manipulations.
Equivalent Rational Expressions and Domain
Consider:
(x²−4)/(x−2)=x+2.
Factor the numerator:
[(x−2)(x+2)]/(x−2)=x+2, for x≠2.
The simplified expression x+2 exists at x=2, but the original expression does not. The two expressions are therefore equivalent only on the original domain x≠2.
Algebraic cancellation can simplify a formula without enlarging its original domain.
Worked Identity 2: Rational Expression
Show, for x≠±1, that:
1/(x−1)−1/(x+1)=2/(x²−1).
Start from the left:
[(x+1)−(x−1)]/[(x−1)(x+1)]
=2/(x²−1).
The proof depends on common-denominator algebra and the difference-of-squares identity.
Do Not Prove an Identity by Manipulating Both Sides Until They Meet
Working independently on both sides can hide circular reasoning. A safer proof transforms one side using known identities until it becomes the other side.
In some longer proofs, transforming both sides into a common third expression can be valid if each chain is independently justified. But students should avoid writing steps that quietly assume the target identity.
Trigonometric Identities Are Algebra Plus Known Relationships
The fundamental relationships include:
- sin²θ+cos²θ=1;
- tanθ=sinθ/cosθ;
- secθ=1/cosθ;
- cosecθ=1/sinθ;
- cotθ=cosθ/sinθ.
Many trig proofs become ordinary algebra after conversion to sine and cosine.
Worked Identity 3: Trigonometric Transformation
Show that, where defined:
(1−cos²θ)/sinθ=sinθ.
Using 1−cos²θ=sin²θ:
(1−cos²θ)/sinθ=sin²θ/sinθ=sinθ.
The original left side requires sinθ≠0. The identity holds over that common domain.
Counterexamples Disprove Universal Claims
To prove an identity, one numerical example is never enough. But to disprove a false universal claim, one counterexample is enough.
Claim:
√(a+b)=√a+√b.
Take a=b=1:
√2≠2.
Therefore the claim is false in general.
One example can illustrate a true rule; one counterexample can destroy a false universal rule.
Reversible and Non-Reversible Transformations
Identity proof usually relies on equivalence-preserving transformations. But some equation-solving moves are only one-way without extra conditions.
- expanding/factorising correctly: reversible;
- adding the same expression to both sides: reversible;
- multiplying by a known non-zero constant: reversible;
- squaring both sides: may introduce extra solutions;
- taking square roots: sign information can be lost;
- multiplying by an expression containing x: may hide excluded zero values.
Knowing whether a transformation preserves equivalence determines whether final candidate checking is necessary.
Worked Example 4: Same Equation, Different Form
Show that solving:
x²−6x+5=0
is equivalent to solving:
(x−3)²=4.
Complete the square:
x²−6x+9−4=0
(x−3)²−4=0
(x−3)²=4.
Both forms produce x=1 or 5. The completed-square form also exposes symmetry around x=3.
Identity Proof as Representation Search
When stuck, ask which form makes the target visible:
- factor to expose cancellation;
- expand to combine terms;
- use a common denominator;
- convert trig functions to sine/cosine;
- use Pythagorean identities;
- rationalise using conjugates;
- complete the square;
- rewrite indices/logarithms using known laws.
The problem is often not “do more algebra” but “choose the representation that reveals the equivalence”.
Common Failure Modes
| Error | Cause | Repair |
|---|---|---|
| checks one number and calls identity proved | example confused with proof | transform symbolically for all allowed values |
| works both sides using target assumption | circular reasoning | transform one side independently |
| cancels terms across addition | factor structure ignored | factor before cancellation |
| drops domain restriction after cancellation | expression simplification confused with function domain | carry original exclusions |
| uses invented trig/log law | memorised pattern overgeneralised | derive from known identities/laws |
| equation transformation introduces extra root unnoticed | non-reversible step unchecked | substitute final candidates into original |
A 45-Minute Identity Session
- 8 minutes: prove four polynomial identities by expansion/factorisation.
- 8 minutes: simplify three rational expressions while preserving domain.
- 10 minutes: prove two trig identities from one side.
- 7 minutes: disprove three false claims using counterexamples.
- 7 minutes: classify equation transformations as reversible or requiring checks.
- 5 minutes: explain which representation exposed each proof.
What Mastery Looks Like
- The learner distinguishes identities from equations.
- The learner transforms one side of a proof without circular reasoning.
- The learner uses factorisation, expansion and common denominators deliberately.
- The learner carries domain restrictions through rational/trig identities.
- The learner can disprove false universal claims with counterexamples.
- The learner recognises when a transformation is not fully reversible.
- The learner sees equivalent forms as different views of the same mathematical object.
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