Hyperbolic functions and identitiesEdexcel International A Level Further Maths: Mind map

Exam questions on Hyperbolic functions and identities

1. A student evaluates hyperbolic functions at $x=\ln3$, using $\sinh x=\frac{e^x-e^{-x}}{2}$ and $\cosh x=\frac{e^x+e^{-x}}{2}$.
   Use the identity $\cosh2x=\cosh^2x+\sinh^2x$ to find the exact value of $\cosh(2\ln3)$.2 marks
2. The function $f$ is defined by $f(x)=\tanh x$ for all real $x$.
   Show that $\tanh x=\frac{e^{2x}-1}{e^{2x}+1}$.2 marks
3. For real $x$, $\sinh x=\frac{e^x-e^{-x}}{2}$ and $\cosh x=\frac{e^x+e^{-x}}{2}$.
   Prove that $\cosh^2x-\sinh^2x=1$.3 marks