Trigonometric method of differences

A question is this type if and only if it uses trigonometric identities to create a telescoping sum involving tan, sin, cos, or arctan functions.

7 questions · Challenging +1.0

4.06b Method of differences: telescoping series
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CAIE Further Paper 1 2021 June Q1
6 marks Standard +0.8
1
  1. Show that $$\tan ( r + 1 ) - \tan r = \frac { \sin 1 } { \cos ( r + 1 ) \cos r }$$ Let \(\mathrm { u } _ { \mathrm { r } } = \frac { 1 } { \cos ( \mathrm { r } + 1 ) \cos \mathrm { r } }\).
  2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } u _ { r }\).
  3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.
Edexcel F2 2022 January Q6
11 marks Challenging +1.2
6. Given that \(A > B > 0\), by letting \(x = \arctan A\) and \(y = \arctan B\)
  1. prove that $$\arctan A - \arctan B = \arctan \left( \frac { A - B } { 1 + A B } \right)$$
  2. Show that when \(A = r + 2\) and \(B = r\) $$\frac { A - B } { 1 + A B } = \frac { 2 } { ( 1 + r ) ^ { 2 } }$$
  3. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \arctan \frac { 2 } { ( 1 + r ) ^ { 2 } } = \arctan ( n + p ) + \arctan ( n + q ) - \arctan 2 - \frac { \pi } { 4 }$$ where \(p\) and \(q\) are integers to be determined.
  4. Hence, making your reasoning clear, determine $$\sum _ { r = 1 } ^ { \infty } \arctan \left( \frac { 2 } { ( 1 + r ) ^ { 2 } } \right)$$ giving the answer in the form \(k \pi - \arctan 2\), where \(k\) is a constant.
CAIE FP1 2019 June Q2
5 marks Challenging +1.2
2 Let \(u _ { n } = \frac { 4 \sin \left( n - \frac { 1 } { 2 } \right) \sin \frac { 1 } { 2 } } { \cos ( 2 n - 1 ) + \cos 1 }\).
  1. Using the formulae for \(\cos P \pm \cos Q\) given in the List of Formulae MF10, show that $$u _ { n } = \frac { 1 } { \cos n } - \frac { 1 } { \cos ( n - 1 ) }$$
  2. Use the method of differences to find \(\sum _ { n = 1 } ^ { N } u _ { n }\).
  3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.
CAIE FP1 2015 June Q4
7 marks Challenging +1.2
4 Use the formula for \(\tan ( A - B )\) in the List of Formulae (MF10) to show that $$\tan ^ { - 1 } ( x + 1 ) - \tan ^ { - 1 } ( x - 1 ) = \tan ^ { - 1 } \left( \frac { 2 } { x ^ { 2 } } \right)$$ Deduce the sum to \(n\) terms of the series $$\tan ^ { - 1 } \left( \frac { 2 } { 1 ^ { 2 } } \right) + \tan ^ { - 1 } \left( \frac { 2 } { 2 ^ { 2 } } \right) + \tan ^ { - 1 } \left( \frac { 2 } { 3 ^ { 2 } } \right) + \ldots .$$
AQA FP2 2006 January Q6
12 marks Challenging +1.2
6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).
    1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$ (2 marks)
    2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
    3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
  2. Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form \(a + \mathrm { i } b\).
OCR Further Pure Core 1 2021 November Q2
8 marks Standard +0.3
You are given that \(\mathrm{f}(x) = \tan^{-1}(1 + x)\).
    1. Find the value of \(\mathrm{f}(0)\). [1]
    2. Determine the value of \(\mathrm{f}'(0)\). [2]
    3. Show that \(\mathrm{f}''(0) = -\frac{1}{2}\). [3]
  2. Hence find the Maclaurin series for \(\mathrm{f}(x)\) up to and including the term in \(x^2\). [2]
Pre-U Pre-U 9795/1 2015 June Q13
10 marks Challenging +1.2
  1. By sketching a suitable triangle, show that \(\tan^{-1} a + \tan^{-1} \left(\frac{1}{a}\right) = \frac{1}{4}\pi\), for \(a > 0\). [1]
  2. Given that \(a\) and \(b\) are positive and less than 1, express \(\tan(\tan^{-1} a \pm \tan^{-1} b)\) in terms of \(a\) and \(b\). [2]
  3. By letting \(a = \frac{1}{n-1}\) and \(b = \frac{1}{n+1}\), use the method of differences to prove that $$\sum_{n=1}^{\infty} \tan^{-1} \left(\frac{2}{n^2}\right) = \frac{3}{4}\pi.$$ [7]