A) \[\frac{63}{65}\]
B) \[\frac{61}{65}\]
C) \[\frac{3}{5}\]
D) \[\frac{5}{13}\]
E) \[\frac{8}{65}\]
Correct Answer: A
Solution :
Given that, \[\alpha ,\beta \in \left( 0,\frac{\pi }{2} \right)\] \[\sin \alpha =\frac{4}{5}\] \[\Rightarrow \] \[\cos \alpha =\frac{3}{5}\] Now, \[\cos (\alpha +\beta )=-\frac{12}{13}\] \[\Rightarrow \] \[\sin (\alpha +\beta )=-\frac{5}{13}\] and \[cos\alpha \text{ }cos\beta -sin\alpha \text{ }sin\beta =-\frac{12}{13}\] \[\Rightarrow \] \[\frac{3}{5}\cos \beta -\frac{4}{5}\sin \beta =-\frac{12}{13}\] ...(i) Now, \[\sin (\alpha +\beta )=\frac{5}{13}\] \[\Rightarrow \] \[\sin \alpha \sin \beta +\cos \alpha \sin \beta =\frac{5}{13}\] \[\Rightarrow \] \[\frac{4}{5}\cos \beta +\frac{3}{5}sin\beta =\frac{5}{13}\] ...(ii) Solving Eqs. (i) and (ii) simultaneously, we get \[\sin \beta =\frac{63}{65}\]You need to login to perform this action.
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