4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1

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Edexcel FP3 Q21
7 marks Standard +0.3
Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of exponentials,
  1. prove that \(\cosh^2 x - \sinh^2 x = 1\), [3]
  2. solve \(\operatorname{cosech} x - 2 \coth x = 2\), giving your answer in the form \(k \ln a\), where \(k\) and \(a\) are integers. [4]
AQA FP2 2011 June Q2
10 marks Standard +0.3
  1. Use the definitions of \(\cosh \theta\) and \(\sinh \theta\) in terms of \(e^\theta\) to show that $$\cosh x \cosh y - \sinh x \sinh y = \cosh(x - y)$$ [4 marks]
  2. It is given that \(x\) satisfies the equation $$\cosh(x - \ln 2) = \sinh x$$
    1. Show that \(\tanh x = \frac{5}{4}\). [4 marks]
    2. Express \(x\) in the form \(\frac{1}{2} \ln a\). [2 marks]
OCR FP2 2009 January Q6
8 marks Standard +0.3
  1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$1 + 2\sinh^2 x = \cosh 2x.$$ [3]
  2. Solve the equation $$\cosh 2x - 5\sinh x = 4,$$ giving your answers in logarithmic form. [5]
OCR FP2 2010 January Q5
8 marks Standard +0.3
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$\cosh^2 x - \sinh^2 x \equiv 1.$$ Deduce that \(1 - \tanh^2 x \equiv \operatorname{sech}^2 x\). [4]
  2. Solve the equation \(2\tanh^2 x - \operatorname{sech} x = 1\), giving your answer(s) in logarithmic form. [4]
AQA Further AS Paper 1 2018 June Q17
4 marks Standard +0.8
Find the exact solution to the equation $$\sinh \theta(\sinh \theta + \cosh \theta) = 1$$ [4 marks]
AQA Further AS Paper 1 2019 June Q9
7 marks Challenging +1.2
  1. Saul is solving the equation $$2\cosh x + \sinh^2 x = 1$$ He writes his steps as follows: $$2\cosh x + \sinh^2 x = 1$$ $$2\cosh x + 1 - \cosh^2 x = 1$$ $$2\cosh x - \cosh^2 x = 0$$ $$\cosh x \neq 0 \therefore 2 - \cosh x = 0$$ $$\cosh x = 2$$ $$x = \pm \cosh^{-1}(2)$$ Identify and explain the error in Saul's method. [2 marks]
  2. Anna is solving the different equation $$\sinh^2(2x) - 2\cosh(2x) = 1$$ and finds the correct answers in the form \(x = \frac{1}{p}\cosh^{-1}(q + \sqrt{r})\), where \(p\), \(q\) and \(r\) are integers. Find the possible values of \(p\), \(q\) and \(r\). Fully justify your answer. [5 marks]
AQA Further Paper 1 2019 June Q6
8 marks Standard +0.8
  1. Show that $$\cosh^3 x + \sinh^3 x = \frac{1}{4}e^{mx} + \frac{3}{4}e^{nx}$$ where \(m\) and \(n\) are integers. [3 marks]
  2. Hence find \(\cosh^6 x - \sinh^6 x\) in the form $$\frac{a \cosh(kx) + b}{8}$$ where \(a\), \(b\) and \(k\) are integers. [5 marks]
AQA Further Paper 1 2023 June Q12
6 marks Standard +0.3
  1. Starting from the identities for \(\sinh 2x\) and \(\cosh 2x\), prove the identity $$\tanh 2x = \frac{2 \tanh x}{1 + \tanh^2 x}$$ [2 marks]
    1. The function f is defined by $$f(x) = \tanh x \quad (x > 0)$$ State the range of f [1 mark]
    2. Use part (a) and part (b)(i) to prove that \(\tanh 2x > \tanh x\) if \(x > 0\) [3 marks]
AQA Further Paper 1 2024 June Q9
8 marks Standard +0.8
  1. It is given that $$p = \ln\left(r + \sqrt{r^2 + 1}\right)$$ Starting from the exponential definition of the sinh function, show that \(\sinh p = r\) [4 marks]
  2. Solve the equation $$\cosh^2 x = 2\sinh x + 16$$ Give your answers in logarithmic form. [4 marks]
AQA Further Paper 1 2024 June Q14
7 marks Challenging +1.2
Solve the differential equation $$\frac{dy}{dx} + y\tanh x = \sinh^3 x$$ given that \(y = 3\) when \(x = \ln 2\) Give your answer in an exact form. [7 marks]
AQA Further Paper 1 Specimen Q2
2 marks Moderate -0.8
Use the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\) to show that \(\cosh^2 x - \sinh^2 x = 1\) [2 marks]
AQA Further Paper 1 Specimen Q10
10 marks Challenging +1.3
The curve, \(C\), has equation \(y = \frac{x}{\cosh x}\)
  1. Show that the \(x\)-coordinates of any stationary points of \(C\) satisfy the equation \(\tanh x = \frac{1}{x}\) [3 marks]
    1. Sketch the graphs of \(y = \tanh x\) and \(y = \frac{1}{x}\) on the axes below. [2 marks]
    2. Hence determine the number of stationary points of the curve \(C\). [1 mark]
  3. Show that \(\frac{d^2y}{dx^2} + y = 0\) at each of the stationary points of the curve \(C\). [4 marks]
AQA Further Paper 1 Specimen Q11
6 marks Standard +0.8
  1. Prove that \(\frac{\sinh \theta}{1 + \cosh \theta} + \frac{1 + \cosh \theta}{\sinh \theta} \equiv 2\coth \theta\) Explicitly state any hyperbolic identities that you use within your proof. [4 marks]
  2. Solve \(\frac{\sinh \theta}{1 + \cosh \theta} + \frac{1 + \cosh \theta}{\sinh \theta} = 4\) giving your answer in an exact form. [2 marks]
OCR MEI Further Pure Core Specimen Q6
6 marks Standard +0.8
  1. A curve is in the first quadrant. It has parametric equations \(x = \cosh t + \sinh t\), \(y = \cosh t - \sinh t\) where \(t \in \mathbb{R}\). Show that the cartesian equation of the curve is \(xy = 1\). [2]
Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the \(x\)-axis, point B lies on the \(y\)-axis and OAPB is a rectangle. \includegraphics{figure_6}
  1. Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer. [4]
SPS SPS FM Pure 2021 May Q8
8 marks Challenging +1.3
  1. Using the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\), show that $$4\sinh^3 x = \sinh 3x - 3\sinh x.$$ [3]
  2. In this question you must show detailed reasoning. By making a suitable substitution, find the real root of the equation $$16u^3 + 12u = 3.$$ Give your answer in the form \(\frac{(a^{\frac{1}{b}} - a^{-\frac{1}{b}})}{c}\) where \(a\), \(b\) and \(c\) are integers. [5]
SPS SPS FM Pure 2026 November Q4
9 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. The curves with equations $$y = \frac{3}{4}\sinh x \text{ and } y = \tanh x + \frac{1}{5}$$ intersect at just one point \(P\)
    1. Use algebra to show that the \(x\) coordinate of \(P\) satisfies the equation $$15e^{4x} - 48e^{3x} + 32e^x - 15 = 0$$ [3]
    2. Show that \(e^x = 3\) is a solution of this equation. [1]
    3. Hence state the exact coordinates of \(P\). [1]
  2. Show that $$\int_{-4}^{0} \frac{e^x}{x^2} dx = e^{-\frac{1}{4}}$$ [4]
OCR Further Pure Core 2 2018 December Q3
6 marks Standard +0.8
In this question you must show detailed reasoning. Solve the equation \(2\cosh^2 x + 5\sinh x - 5 = 0\) giving each answer in the form \(\ln(p + q\sqrt{r})\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. [6]
Pre-U Pre-U 9795/1 2011 June Q4
8 marks Standard +0.8
  1. On a single diagram, sketch the graphs of \(y = \tanh x\) and \(y = \cosh x - 1\), and use your diagram to explain why the equation \(\text{f}(x) = 0\) has exactly two roots, where $$\text{f}(x) = 1 + \tanh x - \cosh x.$$ [3]
  2. The non-zero root of \(\text{f}(x) = 0\) is \(\alpha\).
    1. Verify that \(1 < \alpha < 1.5\). [1]
    2. Taking \(x_1 = 1.25\) as an initial approximation to \(\alpha\), use the Newton-Raphson iterative method to find \(x_3\), giving your answer to 5 decimal places. [4]
Pre-U Pre-U 9795 Specimen Q3
6 marks Standard +0.3
Solve exactly the equation $$5 \cosh x - \sinh x = 7,$$ giving your answers in logarithmic form. [6]