Grade 12 Maths · Trigonometry

Use a double angle identity

Rewrite trig ratios of double angles, for example sin 2A = 2 sin A cos A and cos 2A = 1 − 2 sin²A, choosing the form that helps.

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Question 6.1 · May/June 2025 · Paper 2 · National (DBE) · 4 marks

6.1 Prove that

2 cos^{2} (4 5^{\circ} + x) = 1 - sin 2 x

.(4)

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More past paper questions: use a double angle identity

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3. Question 5.3 · March 2025 · Provincial · 5 marks
5.3 Prove the identity: $\frac{1}{8} (1 - cos 4 x) = sin^{2} x \cdot cos^{2} x$ (5)
Show the memo (5 ticks)Memo
1. ✓
  Write $4 x$ as $2 \times 2 x$ LHS $= \frac{1}{8} [1 - cos (2 \times 2 x)]$
  1 mark · Use a double angle identity
2. ✓
  Expand $cos (2 \times 2 x)$ $= \frac{1}{8} [1 - (1 - 2 sin^{2} 2 x)]$
  1 mark · Use a double angle identity
3. ✓
  Simplify $= \frac{1}{8} [2 sin^{2} 2 x] = \frac{1}{4} sin^{2} 2 x$
  1 mark · Collect like terms
4. ✓
  Expand sin 2x $= \frac{1}{4} (2 sin x cos x)^{2}$
  1 mark · Use a double angle identity
5. ✓
  Final simplification $= \frac{1}{4} (4 sin^{2} x cos^{2} x) = sin^{2} x cos^{2} x = R H S$
  1 mark · Use a double angle identity
4. 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
5. Question 4.1.2 · March 2022 · Provincial · 3 marks
4.1 Given $sin 2 4^{\circ} = m$ and $cos 3 5^{\circ} = n$ . Determine the following in terms of $m$ or $n$ .
4.1.2 $sin 7 0^{\circ}$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Use double angle identity $sin 7 0^{\circ} = sin (2 \times 3 5^{\circ}) = 2 sin 3 5^{\circ} cos 3 5^{\circ}$
  1 mark · Use a double angle identity
2. ✓
  Determine sin 35Since $cos 3 5^{\circ} = n$ , $sin 3 5^{\circ} = 1 - n^{2}$
  1 mark · Use the square identity
3. ✓
  Substitute and solve $2 (1 - n^{2}) (n) = 2 n 1 - n^{2}$
  1 mark · Use a double angle identity
6. Question 7.2 · March 2022 · Provincial · 5 marks
7.2 Calculate, without the use of a calculator, the value of: $\frac{2 s i n ^{2} 22 , 5 ^{\circ} - 1}{4 s i n 22 , 5 ^{\circ} c o s 22 , 5 ^{\circ}}$ (5)
Show the memo (4 ticks)Memo
1. ✓
  Rewrite numerator $2 sin^{2} 22, 5^{\circ} - 1 = - (1 - 2 sin^{2} 22, 5^{\circ}) = - cos (2 \times 22, 5^{\circ}) = - cos 4 5^{\circ}$
  2 marks · Use a double angle identity
2. ✓
  Rewrite denominator $4 sin 22, 5^{\circ} cos 22, 5^{\circ} = 2 (2 sin 22, 5^{\circ} cos 22, 5^{\circ}) = 2 sin 4 5^{\circ}$
  1 mark · Use a double angle identity
3. ✓
  Substitute values $\frac{- \frac{1}{2}}{2 ( \frac{1}{2} )}$
  1 mark · Use special angle values
4. ✓
  Calculate final answer $= - \frac{1}{2}$
  1 mark · Use special angle values

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