Grade 11 Maths · Trigonometry

Write tan as sin over cos

Replace tan x with sin x ÷ cos x, or the other way round, to simplify an expression, for example sin x / (cos x · tan x) = 1.

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Question 5.3 · May/June 2025 · Paper 2 · National (DBE) · 4 marks

5.3 Calculate the value of the following expression without using a calculator:

(tan 9 2^{\circ}) (tan 9 4^{\circ}) (tan 9 6^{\circ}) ... (tan 17 6^{\circ}) (tan 17 8^{\circ})

(4)

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More past paper questions: write tan as sin over cos

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3. Question 2.4.1 · June 2024 · Paper 2 · Provincial · 5 marks
2.4 Given the identity: $sin^{2} α (tan α + \frac{1}{t a n α}) = tan α$
2.4.1 Prove the identity.(5)
Show the memo (5 ticks)Memo
1. ✓
  tan alpha = sin/cosLHS $= sin^{2} α (\frac{s i n α}{c o s α} + \dots)$
  1 mark · Write tan as sin over cos
2. ✓
  1/tan alpha = cos/sin $\dots + \frac{c o s α}{s i n α}$
  1 mark · Write tan as sin over cos
3. ✓
  Add fractions $= sin^{2} α (\frac{s i n ^{2} α + c o s ^{2} α}{s i n α c o s α})$
  1 mark · Add or subtract algebraic fractions
4. ✓
  Apply identity $= sin^{2} α (\frac{1}{s i n α c o s α})$
  $= \frac{s i n α}{c o s α}$
  1 mark · Use the square identity
5. ✓
  Conclude $= tan α = R H S$
  1 mark · Write tan as sin over cos
4. Question 4.5.1 · November 2025 · Paper 2 · Provincial · 3 marks
4.5 Given the identity: $tan β - sin β . cos β = tan β . sin^{2} β$
4.5.1 Prove the above identity.(3)
Show the memo (3 ticks)Memo
1. ✓
  $\frac{s i n β}{c o s β}$ LHS $= tan β - sin β . cos β = \frac{s i n β}{c o s β} - sin β . cos β$
  1 mark · Write tan as sin over cos
2. ✓
  Simplification $= \frac{s i n β - s i n β c o s ^{2} β}{c o s β}$
  1 mark · Add or subtract algebraic fractions
3. ✓
  Common factor $= \frac{s i n β ( 1 - c o s ^{2} β )}{c o s β} = \frac{s i n β}{c o s β} . sin^{2} β = tan β . sin^{2} β =$ RHS
  1 mark · Take out a common factor
5. Question 7.2.1 · March 2024 · Paper 1 · Provincial · 3 marks
7.2 Given: $\frac{4 s i n x c o s x}{2 s i n ^{2} x - 1}$
7.2.1 Simplify $\frac{4 s i n x c o s x}{2 s i n ^{2} x - 1}$ to a single trigonometric ratio.(3)
Show the memo (2 ticks)Memo
1. ✓
  Identify double angle components $\frac{2 ( 2 s i n x c o s x )}{- ( 1 - 2 s i n ^{2} x )}$
  2 marks · Use a double angle identity
2. ✓
  State single ratio $- 2 tan 2 x$
  1 mark · Write tan as sin over cos
6. Question 3.2 · March 2021 · Paper 1 · Provincial · 5 marks
3.2 Without using a calculator, simplify to a single trigonometric ratio: $1 + sin x \cdot tan (18 0^{\circ} - x) \cdot cos (- x)$ (5)
Show the memo (4 ticks)Memo
1. ✓
  Reduce terms $1 + sin x \cdot (- tan x) \cdot cos x$
  2 marks · Use a reduction formula
2. ✓
  $\frac{s i n x}{c o s x}$ $= 1 - sin x \cdot \frac{s i n x}{c o s x} \cdot cos x$
  1 mark · Write tan as sin over cos
3. ✓
  $1 - sin^{2} x$ $= 1 - sin^{2} x$
  1 mark · Write tan as sin over cos
4. ✓
  Apply identity $= cos^{2} x$
  1 mark · Use the square identity

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