Prove identity then find exact value and solve/integrate

Multi-part questions where the first part proves a trigonometric identity, and subsequent parts include finding an exact value at a specific angle, plus solving an equation or evaluating an integral.

3 questions · Standard +0.5

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CAIE P2 2023 June Q6
10 marks Standard +0.3
6
  1. Show that \(4 \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \cos \left( \theta - \frac { 1 } { 3 } \pi \right) \equiv \sqrt { 3 } + 2 \sin 2 \theta\).
  2. Find the exact value of \(4 \sin \frac { 17 } { 24 } \pi \cos \frac { 1 } { 24 } \pi\).
  3. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } 4 \sin \left( 2 x + \frac { 1 } { 3 } \pi \right) \cos \left( 2 x - \frac { 1 } { 3 } \pi \right) \mathrm { d } x\).
CAIE P2 2022 March Q4
7 marks Standard +0.3
4
  1. Show that \(\sin 2 \theta \cot \theta - \cos 2 \theta \equiv 1\).
  2. Hence find the exact value of \(\sin \frac { 1 } { 6 } \pi \cot \frac { 1 } { 12 } \pi\).
  3. Find the smallest positive value of \(\theta\) (in radians) satisfying the equation $$\sin 2 \theta \cot \theta - 3 \cos 2 \theta = 1 .$$
OCR C3 Q3
7 marks Standard +0.8
  1. Prove the identity $$\sqrt{2} \cos (x + 45)° + 2 \cos (x - 30)° \equiv (1 + \sqrt{3}) \cos x°.$$ [4]
  2. Hence, find the exact value of \(\cos 75°\) in terms of surds. [3]