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The value of $\tan \left(2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right)$ is
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Verified Answer
The correct answer is:
$-\frac{7}{17}$
$\quad 2 \tan ^{-1}\left(\frac{1}{5}\right)=\tan ^{-1}\left[\frac{2 \times \frac{1}{5}}{1-\left(\frac{1}{5}\right)^{2}}\right]$
$=\tan ^{-1}\left[\frac{10}{24}\right]$
$=\tan ^{-1}\left(\frac{5}{12}\right)$
Let $\tan \left(2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right)=\mathrm{x}$
$\Rightarrow \quad \tan \left[\tan ^{-1}\left(\frac{5}{12}\right)-\frac{\pi}{4}\right]=\mathrm{x}$
$\Rightarrow \tan ^{-1}\left(\frac{5}{12}\right)-\frac{\pi}{4}=\tan ^{-1} \mathrm{x}$
$\Rightarrow \tan ^{-1}\left(\frac{5}{12}\right)-\tan ^{-1}(1)=\tan ^{-1} \mathrm{x}$
$\Rightarrow \quad \mathrm{x}=\frac{-7 / 12}{17 / 12}=-7 / 17$
$=\tan ^{-1}\left[\frac{10}{24}\right]$
$=\tan ^{-1}\left(\frac{5}{12}\right)$
Let $\tan \left(2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right)=\mathrm{x}$
$\Rightarrow \quad \tan \left[\tan ^{-1}\left(\frac{5}{12}\right)-\frac{\pi}{4}\right]=\mathrm{x}$
$\Rightarrow \tan ^{-1}\left(\frac{5}{12}\right)-\frac{\pi}{4}=\tan ^{-1} \mathrm{x}$
$\Rightarrow \tan ^{-1}\left(\frac{5}{12}\right)-\tan ^{-1}(1)=\tan ^{-1} \mathrm{x}$
$\Rightarrow \quad \mathrm{x}=\frac{-7 / 12}{17 / 12}=-7 / 17$
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