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  4. If cos (α+β) =(3/5), sin(α-β) = (5/13) and 0 < α, β < (π/4) then tan (2α ) is equal to :

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Q. If $\cos \left(\alpha+\beta\right) =\frac{3}{5}, \sin\left(\alpha-\beta\right) = \frac{5}{13} $ and $ 0 < \alpha, \beta < \frac{\pi}{4} $ then $\tan \left(2\alpha \right)$ is equal to :

JEE MainJEE Main 2019Trigonometric Functions

Solution:

$0 < \alpha +\beta < \frac{\pi}{2} $ and $ \frac{-\pi}{4} <\alpha - \beta < \frac{\pi}{4} $
if $\cos \left(\alpha+\beta \right) = \frac{3}{5} $ then $\tan\left(\alpha+\beta\right) = \frac{4}{3}$ and if $ \sin \left(\alpha -\beta\right)=\frac{5}{13}$ then $ \tan\left(\alpha-\beta\right)= \frac{5}{12} $
(since $ \alpha -\beta $ here lies in the first quadrant)
Now $ \tan\left(2\alpha\right)=\tan\left\{\left(\alpha+\beta\right) +\left(\alpha-\beta\right)\right\}$
$ = \frac{\tan\left(\alpha+\beta\right)+ \tan\left(\alpha-\beta\right)}{1- \tan\left(\alpha+\beta\right).\tan\left(\alpha-\beta\right)} = \frac{\frac{4}{3} + \frac{5}{12}}{1- \frac{4}{3} . \frac{5}{12}} = \frac{63}{16} $