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Hyperbolic Function Formula

Trigonometric functions are similar to Hyperbolic functions.  Hyperbolic functions also can be seen in many linear differential equations, for example in the cubic equations, the calculation of angles and distances in hyperbolic geometry are done through this formula. This has importance in electromagnetic theory, heat transfer, and special relativity.

The basic hyperbolic formulas are sinh, cosh, tanh.

\[\large e^{x}=  cosh\;x + sinh\;x\]

\[\large sinh\;x=\frac{e^{x}-e^{-x}}{2}\]

\[\large cosh\;x=\frac{e^{x}+e^{-x}}{2}\]

\[\large tanh\;x= \frac{\large sinh\;x}{\large cosh\;x} =\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\]



Following is the relationship among hyperbolic function :

\[\large tanh\;x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\]

\[\large coth\;x=\frac{1}{tanh\;x}= \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}\]

\[\large sech\;x=\frac{1}{cosh\;x}= \frac{2}{e^{x}+e^{-x}}\]

\[\large csch\;x=\frac{1}{sinh\;x}= \frac{2}{e^{x}-e^{-x}}\]


Solved Examples

Question: Derive addition identities for sinh (x + y) and cos h(x + y)

In the identity: $tanh(x+y)=\frac{tanh\; x + tanh\; y}{1 + tanh\; x \; tanh\; y}$


$\large tanh(x+y)= \large \frac{sinh(x+y)}{cosh(x+y)}$

Since, $\large sinh\; (x+y) =sinh\; x \; cosh\; y+ cosh\;x \; sinh\; y$

$\large cosh\; (x+y) =cosh\; x \; cosh\; y+ sinh\;x \; sinh\; y$

$\large = \frac{sinh\; x\; cosh\; y + sinh\; y\; cosh\; x}{cosh\; x\; cosh\; y\; + sinh\; x\; sinh\; y}$

Dividing numerator and denominator by $\large cosh\; x \; cosh\; y$

$\large = \frac{\frac{\large sinh\;x}{\large cosh\;x}+\frac{\large sinh\;y}{\large cosh\;y}}{1+\frac{\large sinh\;x}{\large cosh\;x}\cdot \frac{\large sinh\;y}{\large cosh\;y}}$

$\large = \frac{tanh\;x+tanh\;y}{1+tanh\;x\;tanh\;y}$

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