1.05p Proof involving trig: functions and identities

88 questions

Sort by: Default | Easiest first | Hardest first
CAIE P1 2020 June Q7
8 marks Standard +0.3
7
  1. Show that \(\frac { \tan \theta } { 1 + \cos \theta } + \frac { \tan \theta } { 1 - \cos \theta } \equiv \frac { 2 } { \sin \theta \cos \theta }\).
  2. Hence solve the equation \(\frac { \tan \theta } { 1 + \cos \theta } + \frac { \tan \theta } { 1 - \cos \theta } = \frac { 6 } { \tan \theta }\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2021 June Q7
5 marks Standard +0.3
7
  1. Prove the identity \(\frac { 1 - 2 \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \equiv 1 - \tan ^ { 2 } \theta\).
  2. Hence solve the equation \(\frac { 1 - 2 \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } = 2 \tan ^ { 4 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2021 June Q10
7 marks Standard +0.3
10
  1. Prove the identity \(\frac { 1 + \sin x } { 1 - \sin x } - \frac { 1 - \sin x } { 1 + \sin x } \equiv \frac { 4 \tan x } { \cos x }\).
  2. Hence solve the equation \(\frac { 1 + \sin x } { 1 - \sin x } - \frac { 1 - \sin x } { 1 + \sin x } = 8 \tan x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
CAIE P1 2021 June Q4
6 marks Standard +0.3
4
  1. Show that the equation $$\frac { \tan x + \sin x } { \tan x - \sin x } = k$$ where \(k\) is a constant, may be expressed as $$\frac { 1 + \cos x } { 1 - \cos x } = k$$
  2. Hence express \(\cos x\) in terms of \(k\).
  3. Hence solve the equation \(\frac { \tan x + \sin x } { \tan x - \sin x } = 4\) for \(- \pi < x < \pi\).
CAIE P1 2022 June Q4
6 marks Standard +0.3
4
  1. Prove the identity \(\frac { \sin ^ { 3 } \theta } { \sin \theta - 1 } - \frac { \sin ^ { 2 } \theta } { 1 + \sin \theta } \equiv - \tan ^ { 2 } \theta \left( 1 + \sin ^ { 2 } \theta \right)\).
  2. Hence solve the equation $$\frac { \sin ^ { 3 } \theta } { \sin \theta - 1 } - \frac { \sin ^ { 2 } \theta } { 1 + \sin \theta } = \tan ^ { 2 } \theta \left( 1 - \sin ^ { 2 } \theta \right)$$ for \(0 < \theta < 2 \pi\).
CAIE P1 2024 March Q4
6 marks Moderate -0.3
4
  1. Prove that \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } \equiv 2 \tan \theta\). \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_63_1569_333_328} .............................................................................................................................................. \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_65_1570_511_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_62_1570_603_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_685_324} \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_776_324} ...................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_76_1572_952_322} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_72_1570_1137_324} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_74_1572_1226_322} \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-05_77_1575_1315_319}
  2. Hence solve the equation \(\frac { ( \sin \theta + \cos \theta ) ^ { 2 } - 1 } { \cos ^ { 2 } \theta } = 5 \tan ^ { 3 } \theta\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P1 2020 November Q7
6 marks Standard +0.3
7
  1. Show that \(\frac { \sin \theta } { 1 - \sin \theta } - \frac { \sin \theta } { 1 + \sin \theta } \equiv 2 \tan ^ { 2 } \theta\).
  2. Hence solve the equation \(\frac { \sin \theta } { 1 - \sin \theta } - \frac { \sin \theta } { 1 + \sin \theta } = 8\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2022 November Q7
7 marks Standard +0.3
7
  1. Prove the identity \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } \equiv \frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 }\).
  2. Hence find the exact solutions of the equation \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } = 2\) for \(0 \leqslant \theta \leqslant \pi\).
CAIE P1 2009 June Q1
3 marks Standard +0.3
1 Prove the identity \(\frac { \sin x } { 1 - \sin x } - \frac { \sin x } { 1 + \sin x } \equiv 2 \tan ^ { 2 } x\).
CAIE P1 2014 June Q5
7 marks Standard +0.3
5
  1. Prove the identity \(\frac { 1 } { \cos \theta } - \frac { \cos \theta } { 1 + \sin \theta } \equiv \tan \theta\).
  2. Solve the equation \(\frac { 1 } { \cos \theta } - \frac { \cos \theta } { 1 + \sin \theta } + 2 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2014 June Q4
6 marks Standard +0.3
4
  1. Prove the identity \(\frac { \tan x + 1 } { \sin x \tan x + \cos x } \equiv \sin x + \cos x\).
  2. Hence solve the equation \(\frac { \tan x + 1 } { \sin x \tan x + \cos x } = 3 \sin x - 2 \cos x\) for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P1 2015 November Q4
7 marks Standard +0.3
4
  1. Prove the identity \(\left( \frac { 1 } { \sin x } - \frac { 1 } { \tan x } \right) ^ { 2 } \equiv \frac { 1 - \cos x } { 1 + \cos x }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \sin x } - \frac { 1 } { \tan x } \right) ^ { 2 } = \frac { 2 } { 5 }\) for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P1 2017 November Q5
7 marks Moderate -0.3
5
  1. Show that the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) may be expressed as \(5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0\).
    [0pt] [3]
  2. Hence solve the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2018 November Q5
6 marks Standard +0.3
5
  1. Show that the equation $$\frac { \cos \theta - 4 } { \sin \theta } - \frac { 4 \sin \theta } { 5 \cos \theta - 2 } = 0$$ may be expressed as \(9 \cos ^ { 2 } \theta - 22 \cos \theta + 4 = 0\).
  2. Hence solve the equation $$\frac { \cos \theta - 4 } { \sin \theta } - \frac { 4 \sin \theta } { 5 \cos \theta - 2 } = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2018 November Q7
7 marks Standard +0.3
7
  1. Show that \(\frac { \tan \theta + 1 } { 1 + \cos \theta } + \frac { \tan \theta - 1 } { 1 - \cos \theta } \equiv \frac { 2 ( \tan \theta - \cos \theta ) } { \sin ^ { 2 } \theta }\).
  2. Hence, showing all necessary working, solve the equation $$\frac { \tan \theta + 1 } { 1 + \cos \theta } + \frac { \tan \theta - 1 } { 1 - \cos \theta } = 0$$ for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P2 2021 November Q7
9 marks Standard +0.3
7
  1. By first expanding \(\cos ( 2 \theta + \theta )\), show that \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
  2. Find the exact value of \(2 \cos ^ { 3 } \left( \frac { 5 } { 18 } \pi \right) - \frac { 3 } { 2 } \cos \left( \frac { 5 } { 18 } \pi \right)\).
  3. Find \(\int \left( 12 \cos ^ { 3 } x - 4 \cos ^ { 3 } 3 x \right) \mathrm { d } x\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2021 November Q7
10 marks Standard +0.8
7
  1. Prove that \(4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \equiv \sqrt { 3 } - \sqrt { 3 } \cos 2 x + \sin 2 x\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 5 } { 6 } \pi } 4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \mathrm { d } x\).
  3. Find the smallest positive value of \(y\) satisfying the equation $$4 \sin ( 2 y ) \sin \left( 2 y + \frac { 1 } { 6 } \pi \right) = \sqrt { 3 } .$$ Give your answer in an exact form.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2004 June Q7
10 marks Standard +0.3
7
  1. By expanding \(\cos ( 2 x + x )\), show that $$\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$$
  2. Hence, or otherwise, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }$$
CAIE P2 2005 June Q7
10 marks Standard +0.3
7
  1. By expanding \(\sin ( 2 x + x )\) and using double-angle formulae, show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$
  2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x d x = \frac { 5 } { 24 }$$
CAIE P2 2006 June Q2
5 marks Moderate -0.3
2
  1. Prove the identity $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos x$$
  2. Hence solve the equation $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) = 1$$ for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P2 2011 June Q8
9 marks Standard +0.8
8
  1. Prove that \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) \equiv 4 \cos 2 \theta\).
  2. Hence
    1. solve for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\) the equation \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) = 3\),
    2. find the exact value of \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } - \sec ^ { 2 } 15 ^ { \circ }\).
CAIE P2 2013 June Q8
9 marks Standard +0.8
8
  1. Prove the identity $$\frac { 1 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } \equiv \operatorname { cosec } x$$
  2. Hence solve the equation $$\frac { 2 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } = 3 \cot ^ { 2 } x - 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2015 June Q6
10 marks Standard +0.3
6
  1. Prove that \(2 \operatorname { cosec } 2 \theta \tan \theta \equiv \sec ^ { 2 } \theta\).
  2. Hence
    1. solve the equation \(2 \operatorname { cosec } 2 \theta \tan \theta = 5\) for \(0 < \theta < \pi\),
    2. find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 2 \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).
CAIE P3 2005 June Q6
8 marks Standard +0.3
6
  1. Prove the identity $$\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3$$
  2. Hence solve the equation $$\cos 4 \theta + 4 \cos 2 \theta = 2$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2011 June Q9
10 marks Standard +0.8
9
  1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3\).
  2. Hence
    1. solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 1\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\),
    2. find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta \mathrm {~d} \theta\).