Write the value of (2 ^ -1 1 5 ). - Vidyadip

Question

Write the value of $\tan \bigg(2 \tan^{-1}\frac{1}{5}\bigg).$

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Answer

Let 2 $\tan^{-1}\frac{1}{5} =\theta$
$\Rightarrow\tan^{-1}\frac{1}{5} =\frac{\theta}{2}\Rightarrow\tan\frac{\theta}{2} = \frac{1}{5}.$
Now, $\tan\bigg(2\tan^{-1}\frac{1}{5}\bigg) = \tan\theta = \frac{2\tan\frac{\theta}{2}}{1-\tan^{2}\frac{\theta}{2}}$
$ = \frac{2\times\frac{1}{5}}{1-\big(\frac{1}{5}\big)^{2}} =\frac{2}{5}\times\frac{25}{24} = \frac{5}{12}.$

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