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Calculate tanh (2x) given tanh x = 3/4

Find the value of \tanh(2x) given that \tanh = \frac{3}{4}.


We recall the following formulas (from Section 6.19, Exercises #7 and #8),

    \begin{align*}  \sinh(2x) &= 2 \sinh x \cosh x \\  \cosh (2x) &= \cosh^2 x + \sinh^2 x. \end{align*}

Then using the definition of hyperbolic tangent we compute

    \begin{align*}  \tanh (2x) &= \frac{\sinh (2x)}{\cosh (2x)} \\[9pt]  &= \frac{2 \sinh x \cosh x}{\cosh^2 x + \sinh^2 x} \\[9pt]  &= \frac{2 \operatorname{coth} x}{\operatorname{coth}^2 x + 1} \\[9pt]  &= \frac{2 \left( \frac{4}{3} \right)}{\left( \frac{4}{3} \right)^2 + 1} \\[9pt]  &= \frac{24}{25}. \end{align*}

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