Further hyperbolic identities and proofsAQA A-Level Further Maths: Subtopic test

1. 1
   Given that $\tanh x=\frac{3}{5}$ and $x>0$.

   (a)

   Find the value of $\operatorname{sech}^2x$.

   [1 mark]

   • A$\frac{9}{25}$
   • B$\frac{16}{25}$
   • C$\frac{34}{25}$
   • D$\frac{4}{5}$

   (b)

   Find the value of $\cosh x$.

   [1 mark]

   • A$\frac{4}{5}$
   • B$\frac{16}{25}$
   • C$\frac{5}{4}$
   • D$\frac{25}{16}$

   (c)

   Find the exact value of $\sinh 2x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Given that $\coth x=\frac{5}{4}$ and $x>0$.

   (a)

   Find the value of $\operatorname{cosech}^2x$.

   [1 mark]

   • A$\frac{41}{16}$
   • B$\frac{3}{4}$
   • C$\frac{16}{9}$
   • D$\frac{9}{16}$

   (b)

   Find the value of $\sinh x$.

   [1 mark]

   • A$\frac{4}{3}$
   • B$\frac{3}{4}$
   • C$\frac{5}{3}$
   • D$\frac{16}{9}$

   (c)

   Find the exact value of $\cosh 2x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   You may assume $\cosh^2x-\sinh^2x=1$ and $\cosh 2x=\cosh^2x+\sinh^2x$.

   (a)

   Show that $\cosh 2x\equiv1+2\sinh^2x$.

   [3 marks]

   (b)

   Hence solve $\cosh 2x-3\sinh x=1$, giving each answer in exact form.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   You may use $\sinh x=\frac{e^x-e^{-x}}{2}$, $\cosh x=\frac{e^x+e^{-x}}{2}$, $\tanh x=\frac{\sinh x}{\cosh x}$, $\cosh^2x-\sinh^2x=1$, $\sinh 2x=2\sinh x\cosh x$ and $\cosh 2x=\cosh^2x+\sinh^2x$.

   (a)

   (i) Prove that $\tanh 2x\equiv\frac{2\tanh x}{1+\tanh^2x}$.
   (ii) Hence find the value of $\tanh x$ for which $\tanh 2x=\frac45$ and $x>0$.

   [6 marks]

   (b)

   (i) Prove that $\frac{\sinh 2x}{1+\cosh 2x}\equiv\tanh x$.
   (ii) Hence show that $\tanh\left(\frac12\ln5\right)=\frac23$.

   [6 marks]

   Total for question 4: 12 marks

End of questions