Triple Angle Formulas/Hyperbolic Tangent

Theorem

$\tanh {3 x} = \dfrac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}$

where $\tanh$ denotes hyperbolic tangent.

$\ds \tanh {3 x}$

$=$

$\ds \frac {\sinh {3 x} } {\cosh {3 x} }$

$\ds $

$=$

$\ds \frac {3 \sinh x + 4 \sinh^3 x} {\cosh {3 x} }$

$\ds $

$=$

$\ds \frac {3 \sinh x + 4 \sinh^3 x} {4 \cosh^3 x - 3 \cosh x}$

$\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 $

$=$

$\ds \frac {\frac {3 \tanh x} {\cosh^2 x} + 4 \tanh^3 x} {4 - \frac {3 \cosh x} {\cosh^3 x} }$

$\ds $

$=$

$\ds \frac {3 \tanh x \sech^2 x + 4 \tanh^3 x} {4 - 3 \sech^2 x}$

Definition 2 of Hyperbolic Secant

$\ds $

$=$

$\ds \frac {3 \tanh x \paren {1 - \tanh^2 x} + 4 \tanh^3 x} {4 - 3 \paren {1 - \tanh^2 x} }$

$\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

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.32$: Multiple Angle Formulas