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O-LevelAdditional MathematicsTrigonometry4037/22 · Oct/Nov 2016 · Q68 Marks

Prove that cosx/(1+tanx) - sinx/(1+cotx) = cosx - sinx.

(i)Prove that cosx/(1+tanx) - sinx/(1+cotx) = cosx - sinx.[4]
(ii)Hence solve the equation cosx/(1+tanx) - sinx/(1+cotx) = 3sinx-4cosx for -180°<x<180°.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.

[M1] (i) cosx/(1+tanx) - sinx/(1+cotx) tanx = sin x/cos x and cotx = cos x/sin x
= cos²x/(cos x+sin x) - sin²x/(cos x+sin x) [M1] Attempt to multiply by cosx and sinx
= (cos²x-sin²x)/(cos x+sin x) [A1]
= (cosx-sinx)(cosx+sinx)/(cosx+sin x) [A1] AG
[M1] (ii) -sinx + cosx = 3sinx - 4cosx equate and collect sinx and cosx oe
[A1] 5cosx = 4sinx
tanx = 5/4
[A1A1] x=51.3°,-128.7° FT from tan x = k

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About this question

Topic: Trigonometry, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source: O-Level Additional Mathematics, Oct/Nov 2016 Paper 2 Variant 2, question 6.

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