A-level · Beta

Trigonometric identities

Build three proofs one step at a time and retain their domain restrictions.

1−cos⁡2xsin⁡x=sin⁡x\frac{1-\cos^2x}{\sin x}=\sin x

Domain: sin x ≠ 0.

  1. 1−cos⁡2xsin⁡x\frac{1-\cos^2x}{\sin x}

Rewrite the numerator using a known identity.

Numerical check at 30°

Left side ≈ 0.5; right side ≈ 0.5.

Angles in the numerical check are in degrees. The symbolic identities apply on their stated domains. Matching numerical examples illustrate an identity but do not prove it.

Predict, test and explain

Step 1 of 3

Predict

Does simplifying an expression restore inputs excluded by its original denominator?

InvestigateWhich inputs are excluded before you simplify?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Does simplifying an expression restore inputs excluded by its original denominator?

1−cos⁡2xsin⁡x\frac{1-\cos^2x}{\sin x}

Hint

Start from one side and justify each transformation.

Reveal answer

An identity holds on the common domain of the original expressions. Cancelling does not make division by zero valid.

sin⁡x(sin⁡x≠0)\sin x\quad(\sin x\ne0)

Build an explanation

Hint 1

Start from one side and justify each transformation.

Hint 2

An identity holds on the common domain of the original expressions. Cancelling does not make division by zero valid.

Hint 3

Explain why the rule is valid, including its assumptions.

Worked example

This example uses fixed values, separate from the diagram controls.

Show (1 − cos²x)/sin x = sin x on its domain.

  1. Use 1 − cos²x = sin²x.
  2. Cancel one factor of sin x.
  3. Retain sin x ≠ 0 from the original denominator.

Answer: sin x, with sin x ≠ 0.

Connect this idea

Watch out: An identity holds on the common domain of the original expressions. Cancelling does not make division by zero valid.

Specification and learning route

AQA E5, E8 · Edexcel Pure 5.5, Pure 5.8

AS core; A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Sine and cosine; Algebraic fractions; Domain restrictions.

AQA specification · Edexcel specification