A) \[\frac{16}{63}\]
B) \[\frac{56}{33}\]
C) \[\frac{28}{33}\]
D) None of these
Correct Answer: B
Solution :
We have \[\cos \,(\alpha +\beta )=\frac{4}{5}\] and \[\sin \,(\alpha -\beta )=\frac{5}{13}\] \[\Rightarrow \,\,\sin \,(\alpha +\beta )=\frac{3}{5}\] and \[\cos \,(\alpha -\beta )=\frac{12}{13}\] \[\Rightarrow \,\,2\alpha ={{\sin }^{-1}}\frac{3}{5}+{{\sin }^{-1}}\frac{5}{13}\] \[={{\sin }^{-1}}\left[ \frac{3}{5}\sqrt{1-\frac{25}{169}}+\frac{5}{13}\sqrt{1-\frac{9}{25}} \right]\] \[\Rightarrow \,\,2\alpha ={{\sin }^{-1}}\,\left( \frac{56}{65} \right)\,\Rightarrow \,\sin \,2\alpha =\frac{56}{65}\] Now, \[\tan \,2\alpha =\frac{\sin \,2\alpha }{\cos \,2\alpha }=\frac{56/65}{33/65}=\frac{56}{33}\].
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