Double Angle Formulas/Hyperbolic Tangent/Proof 2

Theorem

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

Proof

\(\ds \tanh 2 x\)

\(=\)

\(\ds \tanh \left({x + x}\right)\)

\(\ds \)

\(=\)

\(\ds \frac {\tanh x + \tanh x} {1 + \tanh x \tanh x}\)

Hyperbolic Tangent of Sum

\(\ds \)

\(=\)

\(\ds \frac {2 \tanh x} {1 + \tanh^2 x}\)

$\blacksquare$