Triple Angle Formulas/Hyperbolic Tangent
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Theorem
- $\tanh {3 x} = \dfrac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}$
where $\tanh$ denotes hyperbolic tangent.
Proof
\(\ds \tanh {3 x}\) | \(=\) | \(\ds \frac {\sinh {3 x} } {\cosh {3 x} }\) | Definition 2 of Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \sinh x + 4 \sinh^3 x} {\cosh {3 x} }\) | Triple Angle Formula for Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \sinh x + 4 \sinh^3 x} {4 \cosh^3 x - 3 \cosh x}\) | Triple Angle Formula for Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \sinh x + 4 \sinh^3 x} {4 \cosh^3 x - 3 \cosh x} \frac {\paren {1 / \cosh^3 x} } {\paren {1 / \cosh^3 x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac {3 \tanh x} {\cosh^2 x} + 4 \tanh^3 x} {4 - \frac {3 \cosh x} {\cosh^3 x} }\) | Definition 2 of Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \tanh x \sech^2 x + 4 \tanh^3 x} {4 - 3 \sech^2 x}\) | Definition 2 of Hyperbolic Secant | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \tanh x \paren {1 - \tanh^2 x} + 4 \tanh^3 x} {4 - 3 \paren {1 - \tanh^2 x} }\) | Sum of Squares of Hyperbolic Secant and Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \tanh x - 3 \tanh^3 x + 4 \tanh^3 x} {4 - 3 + 3 \tanh^2 x}\) | multipying out | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}\) | gathering terms |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.32$: Multiple Angle Formulas