Proof Addition and subtraction formulas of Hyperbolic Functions

The Mathematician — Sat, 19 Nov 2022 13:00:01 +0000

We will verify the next addition formulas of the hyperbolic functions:

$\cosh(x+y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y)$
$\sinh(x+y) = \sinh(x)\cosh(y) + \cosh(x)\sinh(y)$
$\tanh(x + y) = \frac{\tanh(x) + \tanh(y)}{1 + \tanh(x)\tanh(y)}$ .

We do know that

\begin{align*} \sinh(x) = \frac{e^x - e^{-x}}{2}, \quad \cosh(x) = \frac{e^x + e^{-x}}{2}, \quad \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}. \end{align*}

Proof 1. We first start with

\cosh(x+y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y)

which is a straightforward calculation:

\begin{align*} \cosh(x)\cosh(y) + \sinh(x)\sinh(y) &= \frac{e^x + e^{-x}}{2} \cdot \frac{e^y - e^{-y}}{2} + \frac{e^x - e^{-x}}{2}\cdot \frac{e^y - e^{-y}}{2} \\ &= \frac{1}{4}(e^{x+y} + e^{x - y} + e^{-x + y} + e^{-x-y}) \ + \\ & \ \quad \frac{1}{4}(e^{x+y} - e^{x - y} - e^{-x + y} + e^{-x-y}) \\ &= \frac{1}{2}(e^{x + y} + e^{-x-y}) \\ &= \cosh(x+y) \end{align*}

Proof 2. The second one is as easy as the previous one:

\begin{align*} \sinh(x)\cosh(y) + \cosh(x)\sinh(y) &= \frac{e^x - e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2} + \frac{e^x + e^{-x}}{2}\cdot \frac{e^y - e^{-y}}{2} \\ &= \frac{1}{4}(e^{x+y} + e^{x - y} - e^{-x + y} - e^{-x-y}) \ + \\ & \ \quad \frac{1}{4}(e^{x+y} - e^{x - y} + e^{-x + y} - e^{-x-y}) \\ &= \frac{1}{2}(e^{x + y} - e^{-x-y}) \\ &= \sinh(x+y) \end{align*}

Proof 3. Since we have already proved that the first two points, we could substitute those results in

\tanh(x + y)

too:

\begin{align*} \tanh(x + y) &= \frac{\sinh(x + y)}{\cosh(x + y)} \\ &= \frac{\sinh(x)\cosh(y) + \cosh(x)\sinh(y)}{\cosh(x)\cosh(y) + \sinh(x)\sinh(y)} \\ &= \frac{\sinh(x)\cosh(y) + \cosh(x)\sinh(y)}{\cosh(x)\cosh(y) + \sinh(x)\sinh(y)} \cdot \frac{\frac{1}{\cosh(x)\cosh(y)}}{\frac{1}{\cosh(x)\cosh(y)}} \\ &= \frac{\frac{\sinh(x)\cosh(y)}{\cosh(x)\cosh(y)} + \frac{\cosh(x)\sinh(y)}{\cosh(x)\cosh(y)}}{\frac{\cosh(x)\cosh(y)}{\cosh(x)\cosh(y)} + \frac{\sinh(x)\sinh(y)}{\cosh(x)\cosh(y)}} \\ &= \frac{\frac{\sinh(x)}{\cosh(x)} + \frac{\sinh(y)}{\cosh(y)}}{1 + \frac{\sinh(x)}{\cosh(x)}\cdot\frac{\sinh(y)}{\cosh(y)}} \\ &= \frac{\tanh(x) + \tanh(y)}{1 + \tanh(x)\tanh(y)}. \end{align*}

Therefore, we have verified all the addition formulas of the hyperbolic functions. We also want to verify the next subtraction formulas of the hyperbolic functions:

$\cosh(x-y) = \cosh(x)\cosh(y) - \sinh(x)\sinh(y)$
$\sinh(x-y) = \sinh(x)\cosh(y) + \cosh(x)\sinh(y)$
$\tanh(x - y) = \frac{\tanh(x) - \tanh(y)}{1 - \tanh(x)\tanh(y)}$ .

The steps will be exact the same as for addition, but we need to switch operations.

Proof 1. We first start with

\cosh(x-y) = \cosh(x)\cosh(y) - \sinh(x)\sinh(y)

\begin{align*} \cosh(x)\cosh(y) - \sinh(x)\sinh(y) &= \frac{e^x + e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2} - \frac{e^x - e^{-x}}{2}\cdot \frac{e^y - e^{-y}}{2} \\ &= \frac{1}{4}(e^{x+y} + e^{x - y} + e^{-x + y} + e^{-x-y}) \ - \\ & \ \quad \frac{1}{4}(e^{x+y} - e^{x - y} - e^{-x + y} + e^{-x-y}) \\ &= \frac{1}{2}(e^{x - y} + e^{-(x-y)}) \\ &= \cosh(x - y) \end{align*}

Proof 2. The second one

\sinh(x-y) = \sinh(x)\cosh(y) + \cosh(x)\sinh(y)

\begin{align*} \sinh(x)\cosh(y) - \cosh(x)\sinh(y) &= \frac{e^x - e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2} - \frac{e^x + e^{-x}}{2}\cdot \frac{e^y - e^{-y}}{2} \\ &= \frac{1}{4}(e^{x+y} + e^{x - y} - e^{-x + y} - e^{-x-y}) \ - \\ & \ \quad \frac{1}{4}(e^{x+y} - e^{x - y} + e^{-x + y} - e^{-x-y}) \\ &= \frac{1}{2}(e^{x - y} - e^{-(x-y)}) \\ &= \sinh(x-y) \end{align*}

Proof 3. We will apply the same method as we did before with

\tanh(x - y)

too:

\begin{align*} \tanh(x - y) &= \frac{\sinh(x - y)}{\cosh(x - y)} \\ &= \frac{\sinh(x)\cosh(y) - \cosh(x)\sinh(y)}{\cosh(x)\cosh(y) - \sinh(x)\sinh(y)} \\ &= \frac{\sinh(x)\cosh(y) - \cosh(x)\sinh(y)}{\cosh(x)\cosh(y) - \sinh(x)\sinh(y)} \cdot \frac{\frac{1}{\cosh(x)\cosh(y)}}{\frac{1}{\cosh(x)\cosh(y)}} \\ &= \frac{\frac{\sinh(x)\cosh(y)}{\cosh(x)\cosh(y)} - \frac{\cosh(x)\sinh(y)}{\cosh(x)\cosh(y)}}{\frac{\cosh(x)\cosh(y)}{\cosh(x)\cosh(y)} - \frac{\sinh(x)\sinh(y)}{\cosh(x)\cosh(y)}} \\ &= \frac{\frac{\sinh(x)}{\cosh(x)} - \frac{\sinh(y)}{\cosh(y)}}{1 - \frac{\sinh(x)}{\cosh(x)}\cdot\frac{\sinh(y)}{\cosh(y)}} \\ &= \frac{\tanh(x) - \tanh(y)}{1 - \tanh(x)\tanh(y)}. \end{align*}

Therefore, we have verified all the substraction formulas of the hyperbolic functions.

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Proof Addition and subtraction formulas of Hyperbolic Functions