O-LevelAdditional MathematicsTrigonometry4037/22 · Oct/Nov 2019 · Q69 Marks

Show that tan x / (1 + sec x) + (1 + sec x) / tan x = 2 / sin x

(i)Show that tan x / (1 + sec x) + (1 + sec x) / tan x = 2 / sin x[5]
(ii)Hence solve the equation tan x / (1 + sec x) + (1 + sec x) / tan x = 1 + 3 sin x for 0° ≤ x ≤ 180°.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

M1 Use tan x = sin x / cos x and sec x = 1 / cos x throughout
M1 dep Multiply by cosx
M1 dep Add their fractions correctly and expand (1 + cosx)² correctly
M1 dep Use sin²x + cos²x = 1 and take out a factor of 2.
A1 All correct AG
OR
M1 Add fractions
M1 dep Expand brackets correctly and use 1 + tan²x = sec²x
M1 dep Cancel secx + 1
M1 dep Use tan x = sin x / cos x and sec x = 1 / cos x oe
A1 All correct AG
B1 Obtain three term quadratic. (e.g., 3sin²x + sinx - 2 = 0 oe)
M1 Solve three term quadratic
A1 41.8° awrt
A1 138.2° awrt Mark final answers This mark is not awarded if there are more solutions in the range.

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Topic: Trigonometry, O-Level Additional Mathematics (syllabus 4037). Structured question, 9 marks.
Source: O-Level Additional Mathematics, Oct/Nov 2019 Paper 2 Variant 2, question 6.

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