A-LevelMathematicsTrigonometry9709/21 · May/June 2025 · Q79 Marks

Prove that sin² 2x+4 cos²x cos2x=4cos⁴x.

(a)Prove that sin² 2x+4 cos²x cos2x=4cos⁴x.[3]
(b)Find the set of possible values of the constant k for which the equation sin² 2x+4 cos²x cos2x+5 = k has no real solutions.[2]
(c)Find the exact value of integral from -π/3 to π/3 of sqrt(sin²(t/2)+4cos²(t/2)) cos(t/2) dt.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[M1] Use correct identities to express in terms of sinx and cosx (Allow for 2sin²xcos² x + 4cos²x(cos² x − sin²x), OE).
[A1] Obtain 4sin² x cos² x + 4cos² x(cos² x − sin²x) (OE).
[A1] Confirm 4cos⁴ x.
[B1] Use identity from part (a) to obtain k <5 or k>9.
[B1] Obtain k <5, k>9.
[M1] State or imply integrand is 2cos²½t or 4cos⁴(t/2).
[M1] Use double-angle identity to express in terms of cost.
[A1] Obtain 1+ cost and integrate to obtain t + sint.
[A1] Use limits to obtain π + √3.

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Topic: Trigonometry, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, May/June 2025 Paper 2 Variant 1, question 7.

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