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Q. The numerical value of tan $\bigg[ 2 tan^{-1}\bigg(\frac{1}{5}\bigg)-\frac{\pi}{4}\bigg]$is...

IIT JEEIIT JEE 1984

Solution:

$\bigg[ 2 tan^{-1}\bigg(\frac{1}{5}\bigg)-\frac{\pi}{4}\bigg] =tan\Bigg[tan^{-1}\Bigg(\frac{2.\frac{1}{5}}{1-\frac{1}{25}}\Bigg)-\frac{\pi}{4}\Bigg]$
$ tan\bigg[tan^{-1}\bigg(\frac{5}{12}\bigg)-\frac{\pi}{4}\bigg]$
$ =\frac{tan\bigg[tan^{-1}\bigg(\frac{5}{12}\bigg)\bigg]-tan\bigg(\frac{\pi}{4}\bigg)}{1+tan\bigg[tan^{-1}\bigg(\frac{5}{12}\bigg)\bigg]tan^{\frac{\pi}{4}}}$
$ =\frac{\frac{5}{12}-1}{1+\frac{5}{12}.1}=-\frac{7}{17}$