Double Angle Formulas/Hyperbolic Tangent/Proof 1
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Theorem
- $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$
Proof
\(\ds \tanh 2 x\) | \(=\) | \(\ds \frac {\sinh 2 x} {\cosh 2 x}\) | Definition 2 of Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \cosh x \sinh x} {\cosh^2 x + \sinh^2 x}\) | Double Angle Formula for Hyperbolic Sine and Double Angle Formula for Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac {2 \cosh x \sinh x} {\cosh^2 x} } {\frac {\cosh^2 x + \sinh^2 x} {\cosh^2 x} }\) | dividing top and bottom by $\cosh^2 x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \tanh x} {1 + \tanh^2 x}\) | Definition 2 of Hyperbolic Tangent |
$\blacksquare$