: Edexcel past-paper questions pack
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Trigonometric identities and proofs

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 164

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 40 parts, 164 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · AS Paper 1 · Q8 · 5 marks
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve, for 0 x 90°, the equation 2 3x = 4 cos 2 3x sin giving your answers, in degrees, to one decimal place. (5)
2. June 2025 · Paper 1 · Q14(a) · 3 marks
14. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. (a) Show that sin(x + 30°) + 3 cos(x + 30°) ≡ 2 cos x (3)
3. June 2025 · Paper 1 · Q16(a) · 4 marks
16. y C R x Figure 5 Figure 5 shows a sketch of the curve C with equation y = 0 < x < 2 x - x The region R, shown shaded in Figure 5, is bounded by C, the line with equation x = 1, the x‑axis and the line with equation x = (a) Use the substitution x = 2 sin u to show that the area of R is given by b∫k cosec 2 u du a where a, b and k are constants to be found. (4)
4. June 2025 · Paper 2 · Q15(b) · 5 marks
(b) Using the substitution x = tan theta show that beta . / x x theta theta f d c os 2 d alpha where alpha and beta are constants to be found. (5)
5. June 2024 · AS Paper 1 · Q13(a) · 4 marks
13. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that the equation sin theta (7 sin theta – 4 cos theta ) = 4 can be written as 2 theta – 4 tan theta – 4 = 0 3 tan (4)
6. June 2024 · Paper 1 · Q13(a) · 4 marks
13. (a) Given that a is a positive constant, use the substitution x = a sin 2 theta to show that pi a : : x a x x a theta theta d sin 2 d (4)
7. June 2024 · Paper 2 · Q8(a) · 3 marks
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Prove that / o 2tan sec theta theta theta n , n < cosec theta cosec theta (3)
8. June 2024 · Paper 2 · Q8(b) · 4 marks
(b) Hence solve, for 0 < x < 90°, the equation O cot x x sec x x cosec cosec Give each answer, in degrees, to one decimal place. (4)
9. June 2023 · AS Paper 1 · Q12(a) · 3 marks
12. In this question you must show detailed reasoning. Solutions relying entirely on calculator technology are not acceptable. (a) Show that the equation x x tan = cos can be written as (3)
10. June 2023 · Paper 2 · Q14(a) · 3 marks
14. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that the equation 2 tan theta (8 cos theta + 23 sin 2 theta ) = 8 sin 2 theta (1 + tan 2 theta ) may be written as 2 theta + B cos theta + C) = 0 sin 2 theta (A cos where A, B and C are constants to be found. (3)
11. June 2023 · Paper 2 · Q15 · 3 marks
15. A student attempts to answer the following question: Given that x is an obtuse angle, use algebra to prove by contradiction that sin x – cos x 1 The student starts the proof with: Assume that sin x – cos x < 1 when x is an obtuse angle Þ(sin x – cos x) 2 < 1 Þ... The start of the student’s proof is reprinted below. Complete the proof. (3)
12. June 2023 · Paper 3 (Mechanics) · Q5(b) · 5 marks
2 alpha – 4 tan alpha + 3 = 0 (b) show that tan (5)
13. June 2022 · AS Paper 1 · Q13(a) · 3 marks
13. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that theta cos theta + tan theta ≡ cos theta theta ≠ (2n + 1)90° n ∈ - sin (3)
14. June 2022 · AS Paper 1 · Q13(b) · 5 marks
Given that cos 2x ≠ 0 (b) solve for 0 < x < 90° cos2x + tan 2x = 3 cos 2x giving your answers to one decimal place. (5)
15. June 2022 · Paper 1 · Q14(a) · 4 marks
14. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Given that 2 sin (x - 60°) = cos (x - 30°) show that tan x = 3 3 (4)
16. June 2022 · Paper 2 · Q15(a) · 3 marks
15. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Given that the first three terms of a geometric series are 12 cos theta 5 + 2 sin theta and 6 tan theta (a) show that 4 sin 2 theta - 52 sin theta + 25 = 0 (3)
17. June 2022 · Paper 2 · Q16(b) · 2 marks
(b) Show that all points on C satisfy the equation y = 1 2 + 5 2 (x - 1) (2)
18. June 2022 · Paper 3 (Mechanics) · Q5(b) · 4 marks
The ball reaches a maximum height of 10 m above the ground. (b) Show that U 2 = 1960 (4)
19. Oct 2021 · Paper 1 · Q10(a) · 4 marks
10. In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Given that 1 + cos 2 theta + sin 2 theta ≠ 0 prove that - theta + theta cos2 sin 2 ≡ theta tan theta theta + + cos2 sin 2 (4)
20. Oct 2021 · Paper 1 · Q10(b) · 4 marks
(b) Hence solve, for 0 < x < 180° - x + x cos sin = x sin x x + + cos sin giving your answers to one decimal place where appropriate. (4)
21. Oct 2021 · AS Paper 1 · Q12(i) · 5 marks
12. In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (i) Solve, for 0 < theta 450°, the equation 2 theta = 6 sin theta 5 cos giving your answers to one decimal place. (5)
22. Oct 2021 · AS Paper 1 · Q12(ii)(a) · 2 marks
(ii) (a) A student’s attempt to solve the question “ Solve, for –90° < x < 90°, the equation 3 tan x – 5 sin x = 0 ” is set out below. 3 tan x – 5 sin x = 0 x 3 sin x – 5 sin x = 0 cos 3 sin x – 5 sin x cos x = 0 3 – 5 cos x = 0 cos x = 3 x = 53.1° Identify two errors or omissions made by this student, giving a brief explanation of each. (2)
23. Oct 2020 · AS Paper 1 · Q9(c) · 5 marks
(c) Solve, for 450° theta < 720°, 3 cos theta = 8 tan theta giving your solution to one decimal place. In part (c) you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (5)
24. Oct 2020 · Paper 1 · Q12(a) · 3 marks
12. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that cosec theta - sin theta ≡ cos theta cot theta theta ≠ (180n)° n (3)
25. Oct 2020 · Paper 1 · Q15(a) · 4 marks
15. The curve C has equation pi x2 tan y = 9 0 < y < 2 (a) Show that d y x = - x d x + (4)
26. Oct 2020 · Paper 2 · Q10(a) · 4 marks
10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that cos 3 A ≡ 4 cos3 A - 3 cos A (4)
27. Oct 2020 · Paper 2 · Q10(b) · 4 marks
(b) Hence solve, for -90° x 180°, the equation 1 - cos 3x = sin2 x (4)
28. June 2019 · Paper 1 · Q6(a) · 6 marks
6. (a) Solve, for –180° - ș - 180°, the equation 5 sin 2ș = 9 tan ș giving your answers, where necessary, to one decimal place. [Solutions based entirely on graphical or numerical methods are not acceptable.] (6)
29. June 2019 · AS Paper 1 · Q12(a) · 4 marks
12. (a) Show that 10sin theta - 7cos theta + Ł 4 ± 5 cos ș + 2cos theta (4)
30. June 2019 · AS Paper 1 · Q12(b) · 3 marks
(b) Hence, or otherwise, solve, for 0 - x : 360q, the equation x x sin - cos + = 4 + 3 sin x x + cos (3)
31. June 2019 · Paper 1 · Q14(c) · 3 marks
(c) Show that, for all points (x, y) lying on C, y d = x d a b - x where a and b are constants to be found. (3)
32. June 2019 · Paper 2 · Q4 · 6 marks
4. y C2 C1 x S Figure 2 The curve C1 with parametric equations 2sin t , 0 - t : 2ʌ x = 10௘cos t , y = 4 meets the circle C2 with equation x2 + y2 = 66 at four distinct points as shown in Figure 2. Given that one of these points, S, lies in the 4th quadrant, find the Cartesian coordinates of S. (6)
33. June 2019 · Paper 2 · Q12(a) · 4 marks
12. (a) Prove cos3 theta + sin3 theta theta theta { 2௘cot 2ș ș z (90n)°, n  ] sin cos (4)
34. June 2018 · Paper 1 · Q5 · 5 marks
5. Given that theta pi pi 3sin y = - theta < < theta + theta 2sin 2cos show that y A pi pi d = - theta < < theta + theta d sin 2 where A is a rational constant to be found. (5)
35. June 2018 · AS Paper 1 · Q7(a) · 4 marks
7. In a triangle ABC, side AB has length 10 cm, side AC has length 5 cm, and angle BAC = ș where ș is measured in degrees. The area of triangle ABC is 15 cm2 (a) Find the two possible values of cos ș (4)
36. June 2018 · AS Paper 1 · Q12(a) · 4 marks
12. (a) Show that the equation 4 cos ș í 1 = 2 sin ș tan ș can be written in the form 6 cos2 ș í cos ș í 2 = 0 (4)
37. June 2018 · Paper 2 · Q7(ii) · 5 marks
(ii) Solve, for 0 theta < 360°, the equation 5 sin theta - 5 cos theta = 2 giving your answers to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5)
38. June 2018 · Paper 2 · Q12(a) · 3 marks
12. (a) Prove that n pi 1 - cos 2 theta ≡ tan theta sin 2 theta , theta ≠ (2 + 1) , n ∈ (3)
39. June 2018 · Paper 2 · Q12(b) · 6 marks
pi < x < 2 pi , the equation (b) Hence solve, for -2 (sec2 x - 5)(1 - cos 2x) = 3 tan2 x sin 2x Give any non-exact answer to 3 decimal places where appropriate. (6)
40. June 2018 · Paper 3 · Q10(b) · 9 marks
The point T is at a horizontal distance of 20 m from A and is at a height of 0.75 m above the ground. The ball reaches T without hitting the ground. (b) Find the size of the angle alpha (9)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2025 · AS Paper 1 · Q8
Mark scheme June 2025 Paper 1 Q8
June 2025 · Paper 1 · Q14
Mark scheme June 2025 Paper 1 Q14(a)
June 2025 · Paper 1 · Q16
Mark scheme June 2025 Paper 1 Q16(a)
June 2025 · Paper 2 · Q15
Mark scheme June 2025 Paper 2 Q15(b)
June 2024 · AS Paper 1 · Q13
Mark scheme June 2024 Paper 1 Q13(a)
June 2024 · Paper 1 · Q13
Mark scheme June 2024 Paper 1 Q13(a)
June 2024 · Paper 2 · Q8
Mark scheme June 2024 Paper 2 Q8(b)
June 2023 · AS Paper 1 · Q12
Mark scheme June 2023 Paper 1 Q12(a)
June 2023 · Paper 2 · Q14
Mark scheme June 2023 Paper 2 Q14(a)
June 2023 · Paper 2 · Q15
Mark scheme June 2023 Paper 2 Q15
June 2023 · Paper 3 (Mechanics) · Q5
Mark scheme June 2023 Paper 3 (Mechanics) Q5(b)
June 2022 · AS Paper 1 · Q13
Mark scheme June 2022 Paper 1 Q13(b)
June 2022 · Paper 1 · Q14
Mark scheme June 2022 Paper 1 Q14(a)
June 2022 · Paper 2 · Q15
Mark scheme June 2022 Paper 2 Q15(a)
June 2022 · Paper 2 · Q16
Mark scheme June 2022 Paper 2 Q16(b)
June 2022 · Paper 3 (Mechanics) · Q5
Mark scheme June 2022 Paper 3 (Mechanics) Q5(b)
Oct 2021 · Paper 1 · Q10
Mark scheme Oct 2021 Paper 1 Q10(b)
Oct 2021 · AS Paper 1 · Q12
Mark scheme Oct 2021 Paper 1 Q12(ii)(a)
Oct 2020 · AS Paper 1 · Q9
Mark scheme Oct 2020 Paper 1 Q9(c)
Oct 2020 · Paper 1 · Q12
Mark scheme Oct 2020 Paper 1 Q12(a)
Oct 2020 · Paper 1 · Q15
Mark scheme Oct 2020 Paper 1 Q15(a)
Oct 2020 · Paper 2 · Q10
Mark scheme Oct 2020 Paper 2 Q10(b)
June 2019 · Paper 1 · Q6
Mark scheme June 2019 Paper 1 Q6(a)
June 2019 · AS Paper 1 · Q12
Mark scheme June 2019 Paper 1 Q12(b)
June 2019 · Paper 1 · Q14
Mark scheme June 2019 Paper 1 Q14(c)
June 2019 · Paper 2 · Q4
Mark scheme June 2019 Paper 2 Q4
June 2019 · Paper 2 · Q12
Mark scheme June 2019 Paper 2 Q12(a)
June 2018 · Paper 1 · Q5
Mark scheme June 2018 Paper 1 Q5
June 2018 · AS Paper 1 · Q7
Mark scheme June 2018 Paper 1 Q7(a)
June 2018 · AS Paper 1 · Q12
Mark scheme June 2018 Paper 1 Q12(a)
June 2018 · Paper 2 · Q7
Mark scheme June 2018 Paper 2 Q7(ii)
June 2018 · Paper 2 · Q12
Mark scheme June 2018 Paper 2 Q12(b)
June 2018 · Paper 3 · Q10
Mark scheme June 2018 Paper 3 Q10(b)