A-LevelMathematicsTrigonometry9709/23 · May/June 2018 · Q710 Marks

Show that 2 cosec²2x(1 – cos 2x) = sec²x.

(i)Show that 2 cosec²2x(1 – cos 2x) = sec²x.[3]
(ii)Solve the equation 2 cosec²2x(1 – cos 2x) = tan x + 21 for 0 < x < π, giving your answers correct to 3 significant figures.[4]
(iii)Find ∫ [2 cosec²(4y + 2) – 2 cosec²(4y + 2) cos(4y + 2)] dy.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[B1] Express cosec²2x as 1/(4sin²x cos²x)
[M1] Attempt to express LHS in terms of sinx and cosx only (Must be using correct working for M1)
[A1] Obtain 2sin²x / (4sin²x cos²x) or equivalent and hence sec²x (AG; necessary detail needed)
[B1] Express equation as 1 + tan²x = tan x + 21
[M1] Solve 3-term quadratic equation for tan x
[A1] Obtain tan x = 5 and hence x = 1.37 (Or greater accuracy 1.3734...)
[A1] Obtain tan x = -4 and hence x = 1.82 (Or greater accuracy 1.8157...)
[B1] Use x = 2y+1
[M1] Identify integral as of form ∫ sec²(ay + b) dy (Condone absence of or error with dy)
[A1] Obtain tan(2y + 1) + c

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