A) \[{{\sin }^{-1}}\frac{63}{65}\]
B) \[{{\sin }^{-1}}\frac{12}{13}\]
C) \[{{\sin }^{-1}}\frac{65}{68}\]
D) \[{{\sin }^{-1}}\frac{5}{12}\]
Correct Answer: A
Solution :
Let \[{{\cot }^{-1}}\frac{3}{4}=\theta \,\,\Rightarrow \,\,\cot \theta =\frac{3}{4}\] and \[\sin \theta =\frac{1}{\sqrt{1+{{\cot }^{2}}\theta }}=\frac{1}{\sqrt{1+(9/16)}}=\frac{4}{5}\] Hence \[{{\cot }^{-1}}\frac{3}{4}+{{\sin }^{-1}}\frac{5}{13}={{\sin }^{-1}}\frac{4}{5}+{{\sin }^{-1}}\frac{5}{13}\] \[={{\sin }^{-1}}\left[ \frac{4}{5}.\sqrt{1-\frac{25}{169}}+\frac{5}{13}.\,\sqrt{1-\frac{16}{25}} \right]\] \[={{\sin }^{-1}}\left[ \frac{4}{5}.\frac{12}{13}+\frac{5}{13}.\frac{3}{5} \right]\]\[={{\sin }^{-1}}\left[ \frac{48+15}{65} \right]={{\sin }^{-1}}\frac{63}{65}\].
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