Grade 11 Maths · Trigonometry · Prove a trig identity

Prove a trig identity Grade 11

Show that the left side equals the right side by rewriting one side with the square, quotient, compound or double angle identities and simplifying, for example prove (1 - 2x) 2x = x.

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Grade 11 – How to Prove Trig Identities (Made Simple) | Part 1Mlungisi Nkosi Maths & Science

Proving Trigonometric Identities Grade 12Kevinmathscience

Trigonometry Grade 12: Identity ExamKevinmathscience

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Question 6.1 · November 2025 · Paper 2 · National (DBE) · 6 marks

6.1 Prove that:

[tan (180° - x)] (1 - cos^{2} x) + cos^{2} x = \frac{( s i n x - c o s x ) ( 1 + s i n x . c o s x )}{- c o s x}

(6)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: prove a trig identity

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3. Question 5.4.1 · May/June 2023 · Paper 2 · National (DBE) · 3 marks
5.4 Consider: $\frac{c o s 2 x + s i n 2 x - c o s ^{2} x}{s i n x - 2 c o s x} = - sin x$
5.4.1 Prove the above identity.(3)
Show the memo (3 ticks)Memo
1. ✓
  cos²x − sin²xLHS $= \frac{c o s 2 x + s i n 2 x - c o s ^{2} x}{s i n x - 2 c o s x}$
  $= \frac{c o s ^{2} x - s i n ^{2} x + \dots}{s i n x - 2 c o s x}$
  1 mark · Use a double angle identity
2. ✓
  2 sin x cos x $= \frac{c o s ^{2} x - s i n ^{2} x + 2 s i n x c o s x - c o s ^{2} x}{s i n x - 2 c o s x}$
  $= \frac{- s i n ^{2} x + 2 s i n x c o s x}{s i n x - 2 c o s x}$
  1 mark · Use a double angle identity
3. ✓
  Common factor of −sin x $= \frac{- s i n x ( s i n x - 2 c o s x )}{s i n x - 2 c o s x}$
  $= - sin x$
  $∴$ LHS $=$ RHS
  1 mark · Prove a trig identity
4. Question 5.3.1 · May/June 2022 · Paper 2 · National (DBE) · 3 marks
5.3 Given the identity: $\frac{s i n 3 x}{1 - c o s 3 x} = \frac{1 + c o s 3 x}{s i n 3 x}$
5.3.1 Prove the identity given above.(3)
Show the memo (3 ticks)Memo
1. ✓
  Multiply by "1" $L H S = \frac{s i n 3 x}{1 - c o s 3 x} \times \frac{1 + c o s 3 x}{1 + c o s 3 x}$
  1 mark · Prove a trig identity
2. ✓
  1 − cos² 3x $= \frac{( s i n 3 x ) ( 1 + c o s 3 x )}{( 1 - c o s 3 x ) ( 1 + c o s 3 x )} = \frac{( s i n 3 x ) ( 1 + c o s 3 x )}{1 - c o s ^{2} 3 x}$
  1 mark · Factorise a difference of squares
3. ✓
  Square identity $= \frac{( s i n 3 x ) ( 1 + c o s 3 x )}{s i n ^{2} 3 x} = \frac{1 + c o s 3 x}{s i n 3 x} = R H S$
  1 mark · Use the square identity
5. Question 5.2.1 · November 2023 · Paper 2 · National (DBE) · 4 marks
5.2 Given: $\frac{c o s ^{4} x + s i n ^{2} x . c o s ^{2} x}{1 + s i n x}$
5.2.1 Prove that $\frac{c o s ^{4} x + s i n ^{2} x . c o s ^{2} x}{1 + s i n x} = 1 - sin x$ (4)
Show the memo (4 ticks)Memo
1. ✓
  METHOD 1: FactorsLHS $= \frac{c o s ^{4} x + s i n ^{2} x . c o s ^{2} x}{1 + s i n x}$
  $= \frac{c o s ^{2} x ( c o s ^{2} x + s i n ^{2} x )}{1 + s i n x}$
  1 mark · Take out a common factor
2. ✓
  METHOD 1: $sin^{2} x + cos^{2} x = 1$ $sin^{2} x + cos^{2} x = 1$
  1 mark · Use the square identity
3. ✓
  METHOD 1: $cos^{2} x = 1 - sin^{2} x$ $= \frac{1 - s i n ^{2} x}{1 + s i n x}$
  1 mark · Use the square identity
4. ✓
  METHOD 1: Factors $= \frac{( 1 - s i n x ) ( 1 + s i n x )}{1 + s i n x}$
  $= 1 - sin x$
  $=$ RHS
  1 mark · Prove a trig identity
6. Question 5.2.1 · May/June 2024 · Paper 2 · National (DBE) · 5 marks
5.2 Given: $tan x (1 - cos^{2} x) + cos^{2} x = \frac{( s i n x + c o s x ) ( 1 - s i n x c o s x )}{c o s x}$
5.2.1 Prove the above identity.(5)
Show the memo (5 ticks)Memo
1. ✓
  METHOD 1: $\frac{s i n x}{c o s x}$ LHS $= tan x (1 - cos^{2} x) + cos^{2} x$
  $= \frac{s i n x}{c o s x} (sin^{2} x) + cos^{2} x$
  1 mark · Write tan as sin over cos
2. ✓
  METHOD 1: $sin^{2} x$ $1 - cos^{2} x = sin^{2} x$
  1 mark · Use the square identity
3. ✓
  METHOD 1: Simplification $= \frac{s i n ^{3} x + c o s ^{3} x}{c o s x}$
  1 mark · Add or subtract algebraic fractions
4. ✓
  METHOD 1: Factorisation of cubes $= \frac{( s i n x + c o s x ) ( s i n ^{2} x - s i n x c o s x + c o s ^{2} x )}{c o s x}$
  1 mark · Factorise sum or difference of cubes
5. ✓
  METHOD 1: $sin^{2} x + cos^{2} x = 1$ $= \frac{( s i n x + c o s x ) ( 1 - s i n x c o s x )}{c o s x}$
  $=$ RHS
  1 mark · Prove a trig identity

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