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The value of $\tan \left(\frac{\pi}{8}\right)$ is
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Verified Answer
The correct answer is:
$\sqrt{2}-1$
Since, $\tan \frac{A}{2}=\frac{1-\cos A}{\sin A}$ Putting $\frac{\mathrm{A}}{2}=\frac{\pi}{8}$, we get
$\begin{aligned} \tan \left(\frac{\pi}{8}\right) & =\frac{1-\cos \frac{\pi}{4}}{\sin \frac{\pi}{4}} \\ & =\frac{1-\frac{1}{\sqrt{2}}}{\left(\frac{1}{\sqrt{2}}\right)} \\ \therefore \quad \tan \left(\frac{\pi}{8}\right) & =\sqrt{2}-1\end{aligned}$
$\begin{aligned} \tan \left(\frac{\pi}{8}\right) & =\frac{1-\cos \frac{\pi}{4}}{\sin \frac{\pi}{4}} \\ & =\frac{1-\frac{1}{\sqrt{2}}}{\left(\frac{1}{\sqrt{2}}\right)} \\ \therefore \quad \tan \left(\frac{\pi}{8}\right) & =\sqrt{2}-1\end{aligned}$
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