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Answer: \(\sinh2x=\frac43\), \(\cosh2x=\frac53\).

Formulas: \(\sinh2x=2\sinh x\cosh x=\dfrac{2\tanh x}{1-\tanh^2x}\), \(\cosh2x=\cosh^2x+\sinh^2x=\dfrac{1+\tanh^2x}{1-\tanh^2x}\) (divide numerator and denominator by \(\cosh^2x\) and use \(\mathrm{sech}^2x=1-\tanh^2x\)).

With \(\tanh x=\frac12\): \(1-\tanh^2x=\frac34\).

\(\sinh2x=\dfrac{2\cdot\frac12}{\frac34}=\dfrac43\), \(\cosh2x=\dfrac{1+\frac14}{\frac34}=\dfrac53\).

Check: \(\cosh^22x-\sinh^22x=\frac{25}{9}-\frac{16}{9}=1\). ✓

Doubtly AI · 27 Oct 2026 AI-verified · Oct 2026 Report
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