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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OCR MEI C4 Q5
7 marks Standard +0.3
Show that \(\cot 2\theta = \frac{1 - \tan^2 \theta}{2 \tan \theta}\). Hence solve the equation $$\cot 2\theta = 1 + \tan \theta \quad \text{for } 0° < \theta < 360°.$$ [7]
OCR H240/03 2019 June Q5
9 marks Standard +0.3
In this question you must show detailed reasoning.
  1. Prove that \((\cot \theta + \cosec \theta)^2 = \frac{1 + \cos \theta}{1 - \cos \theta}\). [4]
  2. Hence solve, for \(0 < \theta < 2\pi\), \(3(\cot \theta + \cosec \theta)^2 = 2 \sec \theta\). [5]
AQA Paper 3 2018 June Q8
9 marks Standard +0.3
  1. Prove the identity \(\frac{\sin 2x}{1 + \tan^2 x} = 2\sin x \cos^3 x\) [3 marks]
  2. Hence find \(\int \frac{4\sin 4\theta}{1 + \tan^2 2\theta} d\theta\) [6 marks]
SPS SPS SM Pure 2020 October Q10
7 marks Standard +0.3
  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 $$6\cos^2(\frac{1}{2}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for \(-90° < \theta < 90°\). [3]
SPS SPS FM Pure 2023 September Q7
8 marks Standard +0.8
  1. Prove the identity \(\frac{\cos x}{\sec x + 1} + \frac{\cos x}{\sec x - 1} = 2\cot^2 x\) [3 marks]
  2. Hence, solve the equation $$\frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) + 1} = \cot\left(2\theta + \frac{\pi}{3}\right) - \frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) - 1}$$ in the interval \(0 \leq \theta \leq 2\pi\), giving your values of \(\theta\) to three significant figures where appropriate. [5 marks]
SPS SPS FM Pure 2025 June Q6
9 marks Standard +0.3
  1. Prove that $$1 - \cos 2\theta = \tan \theta \sin 2\theta, \quad \theta \neq \frac{(2n + 1)\pi}{2}, \quad n \in \mathbb{Z}$$ [3]
  2. Hence solve, for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), the equation $$(\sec^2 x - 5)(1 - \cos 2x) = 3\tan^2 x \sin 2x$$ Give any non-exact answer to 3 decimal places where appropriate. [6]
OCR H240/01 2017 Specimen Q8
6 marks Standard +0.3
  1. Show that \(\frac{2\tan\theta}{1 + \tan^2\theta} = \sin 2\theta\). [3]
  1. In this question you must show detailed reasoning. Solve \(\frac{2\tan\theta}{1 + \tan^2\theta} = 3\cos 2\theta\) for \(0 \leq \theta \leq \pi\). [3]
Pre-U Pre-U 9794/1 2011 June Q9
9 marks Standard +0.8
  1. Prove that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\) and deduce that $$\sin \theta + \sin 3\theta = 4 \sin \theta \cos^2 \theta.$$ [5]
  2. Hence find the values of \(\theta\) such that \(0° < \theta < 180°\) that satisfy the equation $$\cot^2 \theta = \sin \theta + \sin 3\theta.$$ [4]