CAIE A-Level Further Math 2.1.3 Hyperbolic functions

Practise proving hyperbolic identities from exponential definitions and applying them within curve and parametric equation work.

Exam points

rewrite sinh, cosh, tanh, sech, coth or cosech in exponential form before proving an identity
combine fractions over a common denominator to prove identities
use a proved hyperbolic identity when simplifying a curve or parametric expression

[Maximum number: 3]

Starting from the definitions of coth and cosech in terms of exponentials, prove that

\operatorname{coth}^{2} x-\operatorname{cosech}^{2} x=1 .

The curve C has equation $y=\ln \operatorname{coth}\left(\frac{1}{2} x\right)$ for x>0.

$\operatorname{coth} x=\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{\mathrm{e}^{x}-\mathrm{e}^{-x}} \quad \operatorname{cosech} x=\frac{2}{\mathrm{e}^{x}-\mathrm{e}^{-x}}$

B1

$\left(\frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{\mathrm{e}^{x}-\mathrm{e}^{-x}}\right)^{2}-\frac{4}{\left(\mathrm{e}^{x}-\mathrm{e}^{-x}\right)^{2}}=\frac{\mathrm{e}^{2 x}+\mathrm{e}^{-2 x}-2}{\left(\mathrm{e}^{x}-\mathrm{e}^{-x}\right)^{2}}=1$

M1 A1

Writes over common denominator, AG.

3

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2.1.3—Hyperbolic functions question 1