Express cos²x or sin²x in terms of cos 2x

A question is this type if and only if it explicitly asks to express cos²x or sin²x in terms of cos 2x (or cos 4x for cos²2x), typically as a preliminary step before integration or solving.

5 questions · Moderate -0.5

1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)
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CAIE P2 2007 June Q6
8 marks Moderate -0.8
6
  1. Express \(\cos ^ { 2 } x\) in terms of \(\cos 2 x\).
  2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 6 } \pi + \frac { 1 } { 8 } \sqrt { } 3$$
  3. By using an appropriate trigonometrical identity, deduce the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 2 } x \mathrm {~d} x .$$
CAIE P2 2009 November Q5
6 marks Moderate -0.3
5
  1. Express \(\cos ^ { 2 } 2 x\) in terms of \(\cos 4 x\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } \cos ^ { 2 } 2 x \mathrm {~d} x\).
CAIE P2 2011 November Q4
6 marks Moderate -0.8
4
  1. Express \(\cos ^ { 2 } x\) in terms of \(\cos 2 x\).
  2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( \cos ^ { 2 } x + \sin 2 x \right) \mathrm { d } x = \frac { 1 } { 8 } \sqrt { } 3 + \frac { 1 } { 12 } \pi + \frac { 1 } { 4 }$$
OCR MEI Paper 1 2018 June Q8
6 marks Standard +0.3
8
  1. Show that \(8 \sin ^ { 2 } x \cos ^ { 2 } x\) can be written as \(1 - \cos 4 x\).
  2. Hence find \(\int \sin ^ { 2 } x \cos ^ { 2 } x \mathrm {~d} x\).
AQA C4 2006 January Q6
7 marks Moderate -0.8
6
  1. Express \(\cos 2 x\) in the form \(a \cos ^ { 2 } x + b\), where \(a\) and \(b\) are constants.
  2. Hence show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } x \mathrm {~d} x = \frac { \pi } { a }\), where \(a\) is an integer.