Two-part show and solve with cos²θ quartic form

A question is this type if and only if part (a) requires showing that a mixed tan²/sin²/cos² equation can be expressed as a quartic in cosθ (i.e. acos⁴x+bcos²x+c=0), and part (b) solves it.

2 questions · Standard +0.6

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CAIE P1 2023 June Q4
7 marks Standard +0.3
4
  1. Show that the equation $$3 \tan ^ { 2 } x - 3 \sin ^ { 2 } x - 4 = 0$$ may be expressed in the form \(a \cos ^ { 4 } x + b \cos ^ { 2 } x + c = 0\), where \(a , b\) and \(c\) are constants to be found.
  2. Hence solve the equation \(3 \tan ^ { 2 } x - 3 \sin ^ { 2 } x - 4 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P3 2017 June Q3
5 marks Standard +0.8
  1. Express the equation \(\cot \theta - 2 \tan \theta = \sin 2\theta\) in the form \(a \cos^4 \theta + b \cos^2 \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be determined. [3]
  2. Hence solve the equation \(\cot \theta - 2 \tan \theta = \sin 2\theta\) for \(90° < \theta < 180°\). [2]