Verifying identities | Year 11

01 / LEARN

The idea, step by step

Prove a statement on its domain

An identity holds for every value where both original sides are defined. Numerical examples can disprove a claim but a finite set of examples cannot prove an identity over all real angles.

Start with the more complicated side and transform it using valid equalities. Useful strategies include writing everything in sine/cosine, taking a common denominator, factorising and using conjugates. Each line should explain why it equals the previous line.

{sin}^{2} x + {cos}^{2} x = 1

Do not lose the exclusions

List denominators and exclude their zeros before simplifying. Cancelling a common factor does not restore inputs excluded by the original expression. The final formula can have a larger natural domain than the expression it replaced.

Avoid assuming the statement and ending with $1 = 1$ unless every step has been justified as reversible. A chain starting from one side and reaching the other is usually clearer and makes cancellation restrictions visible.

A common trap

Restoring cancelled zeros to the domain of the original expression.

Instead: State the exclusions from the original expression even when the simplified formula exists at more points.

02 / WORKED EXAMPLES

See the thinking

Example 1

Prove $tan x + cot x = 1 / (sin x cos x)$ where both sides are defined.

Require $sin x cos x \neq 0$ .
Write the left side as $sin x / cos x + cos x / sin x$ .
Use a common denominator: $({sin}^{2} x + {cos}^{2} x) / (sin x cos x) = 1 / (sin x cos x)$ .

The identity holds wherever neither sine nor cosine is zero.

Example 2

Simplify $(1 - cos 2 x) / sin 2 x$ , with the original restrictions.

Require $sin 2 x \neq 0$ , so $x \neq k π / 2$ .
Replace the numerator by $2 {sin}^{2} x$ and denominator by $2 sin x cos x$ .
Cancel to obtain $tan x$ , retaining the original exclusions.

tan x

, with

x \neq k π / 2

k \in ℤ

Check your understanding

Is $sin x = cos x$ an identity?

Show answer

No.

At $x = 0$ the sides are 0 and 1, so the equality is only true at selected angles.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Prove ${sec}^{2} x - {tan}^{2} x = 1$ wherever defined.
2 marks
Show $sin x / (1 + cos x) = (1 - cos x) / sin x$ where both sides are defined.
2 marks
Prove ${(sin x + cos x)}^{2} = 1 + sin 2 x$ .
2 marks
Simplify ${cos}^{2} x / (1 - sin x)$ , preserving restrictions.
2 marks
Disprove $tan (A + B) = tan A + tan B$ using $A = B = π / 6$ .
2 marks
Prove $(sec x - tan x) (sec x + tan x) = 1$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Prove ${sec}^{2} x - {tan}^{2} x = 1$ wherever defined.
2 marks
For $cos x \neq 0$ , ${tan}^{2} x + 1 = {sec}^{2} x$ .
Divide the Pythagorean identity by ${cos}^{2} x$ and rearrange.
Show $sin x / (1 + cos x) = (1 - cos x) / sin x$ where both sides are defined.
2 marks
On the common domain $sin x \neq 0$ , multiply numerator/denominator by $1 - cos x$ .
Use $1 - {cos}^{2} x = {sin}^{2} x$ .
The original left side alone is also defined at some additional points, such as 0.
It follows by multiplying by the conjugate and cancelling.
Prove ${(sin x + cos x)}^{2} = 1 + sin 2 x$ .
2 marks
The left side is ${sin}^{2} x + {cos}^{2} x + 2 sin x cos x = 1 + sin 2 x$ .
Expand, combine squares and use double-angle sine.
Simplify ${cos}^{2} x / (1 - sin x)$ , preserving restrictions.
2 marks
Factor the numerator as $(1 - sin x) (1 + sin x)$ and cancel the nonzero common factor.
$1 + sin x$ , for $sin x \neq 1$ .
Disprove $tan (A + B) = tan A + tan B$ using $A = B = π / 6$ .
2 marks
Every relevant tangent exists at this pair, so it is a valid counterexample.
The sides are $\sqrt{3}$ and $2 / \sqrt{3}$ , which differ.
Prove $(sec x - tan x) (sec x + tan x) = 1$ .
2 marks
For $cos x \neq 0$ , the product equals ${sec}^{2} x - {tan}^{2} x = 1$ .
Apply difference of squares and the secant identity.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Further Trigonometry: periodic functions and models, trigonometric identities and reciprocal functions; the Library sequence follows the approved school-topic structure.

Official curriculum source ↗

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