Grade 11 Maths · Trigonometry

Use the square identity

Use sin²x + cos²x = 1 to simplify or rewrite an expression, for example 1 − cos²x = sin²x.

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Grade 11 Trigonometry Identities IntroductionMiss Martins Maths and Science

Identities Grade 11: Introduction and practiceKevinmathscience

Identities Grade 11: PracticeKevinmathscience

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

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Question 7.1 · March 2021 · Provincial · 3 marks

7.1 Prove the identity:

\frac{1 - s i n 2 x}{s i n x - c o s x} = sin x - cos x

(3)

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More past paper questions: use the square identity

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3. Question 5.2.2 · November 2024 · Paper 2 · Provincial · 2 marks
5.2 If $cos 49° = k$ , determine the values of the following in terms of $k$ .
5.2.2 $1 - cos^{2} 41°$ (2)
Show the memo (2 ticks)Memo
1. ✓
  METHOD 1: For $sin^{2} 41°$ $1 - cos^{2} 41° = sin^{2} 41°$
  1 mark · Use the square identity
2. ✓
  METHOD 1: Answer $= k^{2}$
  1 mark · Find trig ratios from a sketch
4. Question 3.3.2 · March 2019 · Provincial · 3 marks
3.3 Determine the value of the following in terms of p, if $cos 3 2^{\circ} = p$
3.3.2 $sin (- 32 8^{\circ})$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Reduction $- sin (36 0^{\circ} - 3 2^{\circ})$ or $sin (3 2^{\circ})$
  1 mark · Use a reduction formula
2. ✓
  Identity/Sketch $sin 3 2^{\circ} = 1 - cos^{2} 3 2^{\circ}$
  1 mark · Use the square identity
3. ✓
  Answer $1 - p^{2}$
  1 mark · Use the square identity
5. Question 3.4.1 · March 2025 · Provincial · 5 marks
3.4 Given: $(1 - cos^{2} x) (tan x + \frac{c o s x}{s i n x})$
3.4.1 Prove that $(1 - cos^{2} x) (tan x + \frac{c o s x}{s i n x}) = tan x$ (5)
Show the memo (5 ticks)Memo
1. ✓
  Identify square identity $1 - cos^{2} x = sin^{2} x$
  1 mark · Use the square identity
2. ✓
  Replace $tan x$ by $\frac{s i n x}{c o s x}$ $sin^{2} x (\frac{s i n x}{c o s x} + \frac{c o s x}{s i n x})$
  1 mark · Write tan as sin over cos
3. ✓
  LCD $(cos x sin x)$ $= sin^{2} x (\frac{s i n ^{2} x + c o s ^{2} x}{c o s x s i n x})$
  1 mark · Add or subtract algebraic fractions
4. ✓
  $sin^{2} x + cos^{2} x = 1$ $= sin^{2} x (\frac{1}{c o s x s i n x})$
  1 mark · Use the square identity
5. ✓
  Simplification to $\frac{s i n x}{c o s x}$ $= \frac{s i n x}{c o s x} = tan x = R H S$
  1 mark · Simplify an algebraic fraction
6. Question 4.3.1 · November 2025 · Paper 2 · Provincial · 6 marks
4.3 Given: $\frac{c o s x . t a n x}{2 [ s i n x . c o s ( x - 90° ) + c o s x . c o s ( - x ) ]}$
4.3.1 Simplify the above expression to a single trigonometric ratio of $x$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  $\frac{s i n x}{c o s x}$ $= \frac{c o s x . \frac{s i n x}{c o s x}}{2 [ s i n x . ( \dots ) + c o s x ( \dots ) ]}$
  1 mark · Write tan as sin over cos
2. ✓
  $sin x$ $cos (x - 90°) = cos (90° - x) = sin x$
  1 mark · Use a reduction formula
3. ✓
  $cos x$ $cos (- x) = cos x$
  $= \frac{c o s x . \frac{s i n x}{c o s x}}{2 [ s i n x . ( s i n x ) + c o s x ( c o s x ) ]}$
  1 mark · Use a reduction formula
4. ✓
  Simplification $= \frac{s i n x}{2 [ s i n ^{2} x + c o s ^{2} x ]}$
  1 mark · Simplify an algebraic fraction
5. ✓
  $sin^{2} x + cos^{2} x = 1$ $= \frac{s i n x}{2 ( 1 )}$
  1 mark · Use the square identity
6. ✓
  Answer $= \frac{s i n x}{2}$
  1 mark · Use the square identity

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