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If $x=\log \left[\cot \left(\frac{\pi}{4}+\theta\right)\right]$, then the value of $\sin h x$ is
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Verified Answer
The correct answer is:
$\ -tan 2 \theta$
We know that,
$\begin{aligned}
\sin h x & =\frac{e^x-e^{-x}}{2} \\
& =\frac{\cot \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}+\theta\right)}{2} \\
& =\frac{1-\tan ^2\left(\frac{\pi}{4}+\theta\right)}{2 \tan \left(\frac{\pi}{4}+\theta\right)} \\
& =\frac{1}{\tan 2\left(\frac{\pi}{4}+\theta\right)} \\
& =-\frac{1}{\cot 2 \theta}=-\tan 2 \theta
\end{aligned}$
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