Lesson E8

E8 Proofs involving trigonometry Quiz: AQA Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson E8, Proofs involving trigonometry: 20 multiple choice questions for the AQA Maths (7357), Unit 5: Trigonometry, written with Revision Ninja.

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The 20 questions

  1. In a proof that an identity holds for all theta, what is the standard approach?

    • Show that it holds only for acute angles, then extend it
    • Start from one side and transform it step by step into the other side using known identities
    • Assume the identity is false and derive a contradiction
    • Substitute one value of theta and show both sides agree

  2. Which method is used to disprove a trigonometric statement?

    • Proof by exhaustion over all quadrants
    • A counterexample, one value of theta for which the two sides differ
    • Differentiating both sides with respect to theta
    • Showing both sides are equal when theta = 0 only

  3. Which of these is an identity rather than an equation?

    • tan theta = 1
    • cos 2 theta = 0
    • sin theta = 1/2
    • sin^2 theta + cos^2 theta = 1

  4. What is the key difference between an identity and an equation in trigonometry?

    • Identities contain only sine, while equations contain only cosine
    • An equation holds for all values, while an identity holds only for particular values
    • Identities always use radians, while equations always use degrees
    • An identity holds for all values of the variable, while an equation holds only for particular values

  5. Which identity is used to prove sec^2 theta = 1 + tan^2 theta?

    • tan theta = sin theta / cos theta
    • sin 2 theta = 2 sin theta cos theta
    • sin^2 theta + cos^2 theta = 1
    • cosec^2 theta = 1 + cot^2 theta

  6. Which step is valid when proving a trigonometric identity?

    • Squaring both sides and ignoring the sign of the result
    • Replacing sin theta by its small angle approximation
    • Multiplying both sides by the same non-zero expression
    • Cancelling a factor that may be zero without checking

  7. Which expression is equal to sec theta - cos theta?

    • cos theta tan theta
    • sin theta tan theta
    • sin theta cot theta
    • sin^2 theta + cos theta

  8. Simplify (1 - sin x)(1 + sin x).

    • 1 - 2 sin x
    • cos x
    • sin^2 x
    • cos^2 x

  9. Prove that (1 - cos 2x)/sin 2x simplifies to what?

    • 2 tan x
    • cot x
    • sin x
    • tan x

  10. Using cos 2A = 2 cos^2 A - 1, which expression gives cos^2 A in terms of cos 2A?

    • cos^2 A = cos 2A + 1
    • cos^2 A = 2 cos 2A - 1
    • cos^2 A = (1 + cos 2A)/2
    • cos^2 A = (1 - cos 2A)/2

  11. Simplify sin(A + B)/(cos A cos B).

    • tan A + tan B
    • tan(A + B)
    • tan A tan B
    • sin A + sin B

  12. Expand and simplify (sin x + cos x)^2.

    • 1 + sin 2x
    • 2 + sin 2x
    • 1 - sin 2x
    • 1 + cos 2x

  13. Factorise sin^4 x - cos^4 x fully.

    • (sin x - cos x)^4
    • (sin^2 x - cos^2 x)^2
    • (sin^2 x - cos^2 x)(sin^2 x + cos^2 x)
    • (sin^2 x + cos^2 x)^2

  14. A proof argues: sin^2 theta = cos^2 theta, so sin theta = cos theta for all theta. What is the flaw?

    • The claim is true for every angle theta
    • Taking square roots loses the sign, so sin^2 theta = cos^2 theta only gives sin theta = plus or minus cos theta
    • Squaring is never allowed in any valid proof
    • sin^2 theta and cos^2 theta are never equal

  15. A student says cos x = 1/2 so x = pi/3 for all x. What is the flaw?

    • pi/3 is not a valid angle in radians
    • Only angles below 90 degrees are allowed in proofs
    • cos x = 1/2 is never possible for real x
    • Cosine is periodic and even, so it has several solutions in each period, not only pi/3

  16. Which expression is equal to cos 4x in terms of sin 2x?

    • 1 - 2 sin^2 2x
    • 1 - sin^2 2x
    • sin 4x - 1
    • 2 sin^2 2x - 1

  17. If tan x = t, which expression gives sin 2x in terms of t?

    • 2t/(1 - t^2)
    • 2t
    • 2t/(1 + t^2)
    • t^2/(1 + t^2)

  18. Simplify sin theta/(1 + cos theta) + (1 + cos theta)/sin theta.

    • 2 sec theta
    • 2 cosec theta
    • cosec theta
    • 2 sin theta

  19. Simplify (cos 2x)/(cos x - sin x).

    • 1 + sin 2x
    • cos x - sin x
    • cos x + sin x
    • cos 2x + sin x

  20. Simplify (sec theta + tan theta)(sec theta - tan theta).

    • sec^2 theta + tan^2 theta
    • sec theta tan theta
    • 1
    • 2 sec theta

All AQA Maths quizzes