Quadruple Angle Formulas/Hyperbolic Tangent
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Theorem
- $\tanh 4 x = \dfrac {4 \tanh x + 4 \tanh^3 x} {1 + 6 \tanh^2 x + \tanh^4 x}$
where $\tanh$ denotes hyperbolic tangent.
Proof
| \(\ds \tanh 4 x)\) | \(=\) | \(\ds \frac {\sinh 4 x} {\cosh 4 x}\) | Definition 2 of Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {8 \sinh^3 x \cosh x + 4 \sinh x \cosh x} {\cosh 4 x}\) | Quadruple Angle Formula for Hyperbolic Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {8 \sinh^3 x \cosh x + 4 \sinh x \cosh x} {8 \cosh^4 x - 8 \cosh^2 x + 1}\) | Quadruple Angle Formula for Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {8 \tanh^3 x + 4 \frac {\tanh x} {\cosh^2 x} } {8 - \frac 8 {\cosh^2 x} + \frac 1 {\cosh^4 x} }\) | dividing top and bottom by $\cosh^4 x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {8 \tanh^3 x + 4 \tanh x \sech^2 x} {8 - 8 \sech^2 x + \sech^4 x}\) | Definition 2 of Hyperbolic Secant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {8 \tanh^3 x + 4 \tanh x \paren {1 - \tanh^2 x} } {8 - 8 \paren {1 - \tanh^2 x} + \paren {1 - \tanh^2 x}^2}\) | Sum of Squares of Hyperbolic Secant and Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {4 \tanh^3 x + 4 \tanh x} {1 + 6 \tanh^2 x + \tanh^4 x}\) | multiplying out and gathering terms |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.35$: Multiple Angle Formulas