If \(5\sinh x-\cosh x=5\), find \(\tanh x\).

Q.1 (c) If \(5\sinh x-\cosh x=5\), find \(\tanh x\).

Doubtly Asked 25 Oct 2026 mu engineering Save WhatsApp Report

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Best answer

\(5\cdot\frac{e^x-e^{-x}}2-\frac{e^x+e^{-x}}2=5\Rightarrow4e^x-6e^{-x}=10\).

Multiply by \(e^x\) and put \(e^x=t\): \(4t^2-10t-6=0\Rightarrow2t^2-5t-3=0\Rightarrow(2t+1)(t-3)=0\).

\(t=e^x>0\Rightarrow e^x=3\).

\(\tanh x=\frac{e^{2x}-1}{e^{2x}+1}=\frac{9-1}{9+1}\Rightarrow\) \(\tanh x=\frac45=0.8\).

Check: \(\sinh x=\frac43,\ \cosh x=\frac53\Rightarrow5\cdot\frac43-\frac53=5\) ✓.

Doubtly AI · 25 Oct 2026 AI-verified · Oct 2026 Report

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