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  4. The value of tan [2 tan-1(1/5)-(π/4)] is

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Q. The value of $\tan \left[2\,\tan^{-1}\frac{1}{5}-\frac{\pi}{4}\right]$ is

Inverse Trigonometric Functions

Solution:

$2\,tan^{-1} \frac{1}{5} = tan^{-1} \frac{1}{5} +tan^{-1} \frac{1}{5}$
$ = tan^{-1} \frac{\frac{1}{5} +\frac{1}{5}}{1-\frac{1}{5}\cdot\frac{1}{5}} $
$= tan^{-1} \frac{\frac{2}{5}}{\frac{24}{25}}$
$ = tan^{-1} \frac{5}{12} $
$ tan \left(2tan^{-1} \frac{1}{5} -\frac{\pi}{4}\right) = tan \left(tan^{-1} \frac{5}{12} -\frac{\pi}{4}\right) $
$=\frac{ tan \left(tan^{-1} \frac{5}{12}\right) - tan \frac{\pi}{4}}{1+tan\left(tan^{-1} \frac{5}{12}\right) tan \frac{\pi}{4}} $
$ =\frac{ \frac{5}{12}-1}{1+\frac{5}{12}\cdot1} $
$ = -\frac{\frac{7}{12}}{\frac{17}{12}} $
$ = -\frac{7}{17}$