Solve using sech/tanh identities

1. Solve the equation

$$4 \tanh x - \operatorname { sech } x = 1$$ giving your answer in the form \(x = \ln k\) where \(k\) is a fully simplified rational number.
(6)

1. Solve the equation

$$15 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 13$$ Give your answers in terms of simplified natural logarithms.

1. Find the exact values of x for which

$$4 \tanh ^ { 2 } x - 2 \operatorname { sech } ^ { 2 } x = 3 ,$$ giving your answers in the form \(\pm \ln \mathrm { a }\), where a is real.

1. Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\) where \(a\) is a rational number.

1. Solve the equation

$$5 \tanh x + 7 = 5 \operatorname { sech } x$$ Give each answer in the form \(\ln k\) where \(k\) is a rational number.

1. Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\), where \(a\) is a rational number.

4

1. Sketch the graph of \(y = \tanh x\).
2. Given that \(u = \tanh x\), use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that $$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
3. 1. Show that the equation $$3 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 5$$ can be written as $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$
   2. Show that the equation $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$ has only one solution for \(x\).
      Find this solution in the form \(\frac { 1 } { 2 } \ln a\), where \(a\) is an integer.

2

1. Sketch the graph of \(y = \tanh x\) and state the equations of its asymptotes.
2. Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that $$\operatorname { sech } ^ { 2 } x + \tanh ^ { 2 } x = 1$$
3. Solve the equation \(6 \operatorname { sech } ^ { 2 } x = 4 + \tanh x\), giving your answers in terms of natural logarithms.
   [0pt] [5 marks] \section*{Answer space for question 2}
   1. \includegraphics[max width=\textwidth, alt={}, center]{bc3aaed2-4aef-4aec-b657-098b1e581e55-04_855_1447_920_324}

1. Show that, for \(0 < x \leq 1\), $$\ln \left(\frac{1 - \sqrt{1-x^2}}{x}\right) = -\ln \left(\frac{1 + \sqrt{1-x^2}}{x}\right).$$ [3]
2. Using the definition of \(\cosh x\) or \(\operatorname{sech} x\) in terms of exponentials, show that, for \(0 < x \leq 1\), $$\operatorname{arsech} x = \ln \left(\frac{1 + \sqrt{1-x^2}}{x}\right).$$ [5]
3. Solve the equation $$3 \tanh^2 x - 4 \operatorname{sech} x + 1 = 0,$$ giving exact answers in terms of natural logarithms. [5]

(Total 13 marks)

Which one of these functions has the set \(\{x : |x| < 1\}\) as its greatest possible domain? Circle your answer. [1 mark] \(\cosh x\) \quad \(\cosh^{-1} x\) \quad \(\tanh x\) \quad \(\tanh^{-1} x\)

Show that the solutions to the equation $$3\tanh^2 x - 2\operatorname{sech} x = 2$$ can be expressed in the form $$x = \pm \ln(a + \sqrt{b})$$ where \(a\) and \(b\) are integers to be found. You may use without proof the result \(\cosh^{-1} y = \ln(y + \sqrt{y^2 - 1})\) [5 marks]