| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Prove hyperbolic identity from exponentials |
| Difficulty | Standard +0.3 Part (a) is a standard bookwork proof requiring direct substitution of exponential definitions—routine for Further Maths students. Part (b) involves algebraic manipulation of hyperbolic functions and solving a quadratic in e^x, which is a typical FP3 exercise requiring multiple steps but following standard techniques without novel insight. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
Using the definitions of $\cosh x$ and $\sinh x$ in terms of exponentials,
\begin{enumerate}[label=(\alph*)]
\item prove that $\cosh^2 x - \sinh^2 x = 1$, [3]
\item solve $\operatorname{cosech} x - 2 \coth x = 2$,
giving your answer in the form $k \ln a$, where $k$ and $a$ are integers. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q21 [7]}}