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If $\cosh 2 x=199$, then coth $x$ equals
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1974 Upvotes
Verified Answer
The correct answer is:
$\frac{10}{3 \sqrt{11}}$
We have,
$$
\begin{aligned}
& \cosh 2 x & =199 \\
\Rightarrow & \frac{1+\tanh ^2 x}{1-\tanh ^2 x} & =199 \\
\Rightarrow & 1+\tanh ^2 x & =199-199 \tanh ^2 x \\
\Rightarrow & 200 \tanh ^2 x & =198 \\
\Rightarrow & \quad \tanh ^2 x & =\frac{198}{200} \Rightarrow \tanh ^2 x=\frac{99}{100} \\
\Rightarrow & & \tanh x=\frac{3 \sqrt{11}}{10} \Rightarrow \operatorname{coth} x=\frac{10}{3 \sqrt{11}}
\end{aligned}
$$
$$
\begin{aligned}
& \cosh 2 x & =199 \\
\Rightarrow & \frac{1+\tanh ^2 x}{1-\tanh ^2 x} & =199 \\
\Rightarrow & 1+\tanh ^2 x & =199-199 \tanh ^2 x \\
\Rightarrow & 200 \tanh ^2 x & =198 \\
\Rightarrow & \quad \tanh ^2 x & =\frac{198}{200} \Rightarrow \tanh ^2 x=\frac{99}{100} \\
\Rightarrow & & \tanh x=\frac{3 \sqrt{11}}{10} \Rightarrow \operatorname{coth} x=\frac{10}{3 \sqrt{11}}
\end{aligned}
$$
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