Solve equation using proven identity

A question is this type if and only if it asks to solve a trigonometric equation by first proving an identity and then using that result (typically marked as 'hence').

33 questions · Standard +0.3

1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals
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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 \(0 \leqslant \theta \leqslant \pi\).
Edexcel PURE 2024 October Q5
Challenging +1.2
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Show that \(\sin 3 x\) can be written in the form
    $$P \sin x + Q \sin ^ { 3 } x$$ where \(P\) and \(Q\) are constants to be found.
  2. Hence or otherwise, solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), the equation $$2 \sin 3 \theta = 5 \sin 2 \theta$$ giving your answers, in degrees, to one decimal place as appropriate.
Pre-U Pre-U 9794/1 2012 June Q10
9 marks Standard +0.3
10
  1. Prove that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta \right)$$ and hence show that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \cos 2 \theta$$
  2. Hence solve the equation \(\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 2\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
Pre-U Pre-U 9794/2 2013 June Q9
12 marks Challenging +1.2
9
  1. Prove that \(\operatorname { cosec } 2 x - \cot 2 x \equiv \tan x\) and hence find an exact value for \(\tan \left( \frac { 3 } { 8 } \pi \right)\).
  2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 3 } { 8 } \pi } ( \operatorname { cosec } 2 x - \cot 2 x ) ^ { 2 } \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2016 June Q11
5 marks Challenging +1.2
11
  1. Prove that $$\sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \frac { 1 } { 2 } \sin ^ { 2 } \theta - \frac { 3 } { 4 } = \frac { 1 } { 4 } \sqrt { 3 } \sin 2 \theta .$$
  2. Hence solve the equation $$2 \sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \sin ^ { 2 } \theta = 1 \text { for } - \pi \leqslant \theta \leqslant \pi .$$
Pre-U Pre-U 9794/1 2016 Specimen Q10
8 marks Standard +0.3
10
  1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
  2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
Pre-U Pre-U 9794/1 2020 Specimen Q10
4 marks Standard +0.3
10
  1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
  2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
CAIE P1 2024 June Q5
6 marks Moderate -0.8
  1. Prove the identity \(\frac{\sin^2 x - \cos x - 1}{1 + \cos x} \equiv -\cos x\). [3]
  2. Hence solve the equation \(\frac{\sin^2 x - \cos x - 1}{2 + 2\cos x} = \frac{1}{4}\) for \(0° \leq x \leq 360°\). [3]
CAIE P1 2023 November Q7
9 marks Standard +0.3
  1. Verify the identity \((2x - 1)(4x^2 + 2x - 1) \equiv 8x^3 - 4x + 1\). [1]
  2. Prove the identity \(\frac{\tan^2\theta + 1}{\tan^2\theta - 1} \equiv \frac{1}{1 - 2\cos^2\theta}\). [3]
  3. Using the results of (a) and (b), solve the equation $$\frac{\tan^2\theta + 1}{\tan^2\theta - 1} = 4\cos\theta,$$ for \(0° < \theta \leqslant 180°\). [5]
CAIE P1 2011 June Q5
6 marks Moderate -0.3
  1. Prove the identity \(\frac{\cos \theta}{\tan \theta(1 - \sin \theta)} \equiv 1 + \frac{1}{\sin \theta}\). [3]
  2. Hence solve the equation \(\frac{\cos \theta}{\tan \theta(1 - \sin \theta)} = 4\), for \(0° \leq \theta \leq 360°\). [3]
CAIE P1 2012 June Q5
6 marks Moderate -0.3
  1. Prove the identity \(\tan x + \frac{1}{\tan x} = \frac{1}{\sin x \cos x}\). [2]
  2. Solve the equation \(\frac{2}{\sin x \cos x} = 1 + 3 \tan x\), for \(0° \leqslant x \leqslant 180°\). [4]
CAIE P1 2015 June Q5
5 marks Moderate -0.3
  1. Prove the identity \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} \equiv \frac{\tan \theta - 1}{\tan \theta + 1}\). [1]
  2. Hence solve the equation \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = \frac{\tan \theta}{6}\), for \(0° \leqslant \theta \leqslant 180°\). [4]
CAIE P1 2017 June Q5
6 marks Moderate -0.3
  1. Show that the equation \(\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta\) may be expressed as \(\cos^2 \theta = 2 \sin^2 \theta\). [3]
  2. Hence solve the equation \(\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta\) for \(0° < \theta < 180°\). [3]
CAIE P1 2014 November Q5
7 marks Moderate -0.3
  1. Show that \(\sin^2 \theta - \cos^4 \theta = 2 \sin^2 \theta - 1\). [3]
  2. Hence solve the equation \(\sin^2 \theta - \cos^4 \theta = \frac{1}{2}\) for \(0° \leq \theta \leq 360°\). [4]
CAIE P2 2024 June Q7
10 marks Standard +0.3
  1. Prove that \(2\sin\theta\cosec 2\theta \equiv \sec\theta\). [2]
  2. Solve the equation \(\tan^2\theta + 7\sin\theta\cosec 2\theta = 8\) for \(-\pi < \theta < \pi\). [5]
  3. Find \(\int 8\sin^2\frac{1}{2}x\cosec^2 x \, dx\). [3]
CAIE P2 2024 November Q7
11 marks Standard +0.8
  1. Prove that \(\cos(\theta + 30°)\cos(\theta + 60°) = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin 2\theta\). [4]
  2. Solve the equation \(5\cos(2\alpha + 30°)\cos(2\alpha + 60°) = 1\) for \(0° < \alpha < 90°\). [4]
  3. Show that the exact value of \(\cos 20° \cos 50° + \cos 40° \cos 70°\) is \(\frac{1}{2}\sqrt{3}\). [3]
CAIE P3 2017 June Q7
9 marks Standard +0.3
  1. Prove that if \(y = \frac{1}{\cos \theta}\) then \(\frac{dy}{d\theta} = \sec \theta \tan \theta\). [2]
  2. Prove the identity \(\frac{1 + \sin \theta}{1 - \sin \theta} = 2 \sec^2 \theta + 2 \sec \theta \tan \theta - 1\). [3]
  3. Hence find the exact value of \(\int_0^{\frac{\pi}{4}} \frac{1 + \sin \theta}{1 - \sin \theta} d\theta\). [4]
CAIE P3 2018 June Q7
9 marks Moderate -0.3
    1. Express \(\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1}\) in the form \(a \sin^2 \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
    2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1} = \frac{1}{4}$$ for \(-90° \leqslant \theta \leqslant 0°\). [2]
  2. \includegraphics{figure_7b} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(-\pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
    1. Find the \(x\)-coordinate of \(A\). [2]
    2. Find the \(y\)-coordinate of \(B\). [2]
CAIE P3 2013 November Q5
7 marks Standard +0.3
  1. Prove that \(\cot \theta + \tan \theta = 2\cosec 2\theta\). [3]
  2. Hence show that \(\int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} \cosec 2\theta \, d\theta = \frac{1}{2}\ln 3\). [4]
Edexcel C3 Q8
9 marks Standard +0.3
  1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan \theta, \quad \theta \neq \frac{n\pi}{2}, \quad n \in \mathbb{Z}.$$ [3]
  2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]
Edexcel C3 Q6
9 marks Standard +0.3
  1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} \equiv \tan \theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [3]
  2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]
OCR C3 Q7
9 marks Standard +0.3
  1. Write down the formula for \(\cos 2x\) in terms of \(\cos x\). [1]
  2. Prove the identity \(\frac{4 \cos 2x}{1 + \cos 2x} = 4 - 2 \sec^2 x\). [3]
  3. Solve, for \(0 < x < 2\pi\), the equation \(\frac{4 \cos 2x}{1 + \cos 2x} = 3 \tan x - 7\). [5]
OCR C3 Q9
12 marks Standard +0.8
  1. Prove the identity $$\tan(\theta + 60°) \tan(\theta - 60°) \equiv \frac{\tan^2 \theta - 3}{1 - 3 \tan^2 \theta}.$$ [4]
  2. Solve, for \(0° < \theta < 180°\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = 4 \sec^2 \theta - 3,$$ giving your answers correct to the nearest \(0.1°\). [5]
  3. Show that, for all values of the constant \(k\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = k^2$$ has two roots in the interval \(0° < \theta < 180°\). [3]
OCR C3 2013 January Q9
10 marks Standard +0.8
  1. Prove that $$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
  2. Hence solve the equation $$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for \(-90° < \theta < 90°\). [3]
  3. It is given that there are two values of \(\theta\), where \(-90° < \theta < 90°\), satisfying the equation $$6\cos^2(\frac{1}{3}\theta + 45°) - 3(\cos \frac{2}{3}\theta - \sin \frac{2}{3}\theta) = k,$$ where \(k\) is a constant. Find the set of possible values of \(k\). [3]
OCR C4 Q7
8 marks Standard +0.3
  1. Show that \(\cos(\alpha + \beta) = \frac{1 - \tan\alpha\tan\beta}{\sec\alpha\sec\beta}\). [3]
  2. Hence show that \(\cos 2\alpha = \frac{1 - \tan^2\alpha}{1 + \tan^2\alpha}\). [2]
  3. Hence or otherwise solve the equation \(\frac{1 - \tan^2\theta}{1 + \tan^2\theta} = \frac{1}{2}\) for \(0° \leqslant \theta \leqslant 180°\). [3]