1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2

306 questions

Sort by: Default | Easiest first | Hardest first
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 Q4
5 marks Standard +0.8
Given that $$y = \frac{3\sin \theta}{2\sin \theta + 2\cos \theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ show that $$\frac{dy}{d\theta} = \frac{A}{1 + \sin 2\theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ where \(A\) is a rational constant to be found. [5]
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]
SPS SPS FM 2025 October Q13
8 marks Challenging +1.8
In this question you must show detailed reasoning. Solve the following equation for \(x\) in the interval \(0° < x < 180°\) $$1 + \log_3\left(1 + \tan^2 2x\right) = 2\log_3(-4\sin 2x)$$ [8]
Pre-U Pre-U 9794/2 2016 June Q10
10 marks Challenging +1.2
  1. Using the substitution \(u = \frac{1}{x}\), or otherwise, find \(\int \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [4]
  2. Evaluate \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) and \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [3]
  3. Show that, when \(n\) is a positive integer, the integral \(\int_{\frac{1}{(n+1)\pi}}^{\frac{1}{n\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) takes one of the two values found in part (ii), distinguishing between the two cases. [3]
Edexcel AEA 2015 June Q3
9 marks Challenging +1.8
Solve for \(0 < x < 360°\) $$\cot 2x - \tan 78° = \frac{(\sec x)(\sec 78°)}{2}$$ where \(x\) is not an integer multiple of \(90°\) [9]