A) \[\frac{1}{\sqrt{10}}\]
B) \[-\frac{1}{\sqrt{10}}\]
C) \[\frac{1}{10}\].
D) \[-\frac{1}{10}\]
Correct Answer: A
Solution :
Let \[{{\cos }^{-1}}\frac{4}{5}=x\] Þ \[\cos x=\frac{4}{5}\] .....(i) Now \[\sin \left( \frac{1}{2}{{\cos }^{-1}}\frac{4}{5} \right)=\sin \left( \frac{x}{2} \right)\] .....(ii) From (i), \[\cos x=\frac{4}{5}\]Þ \[1-2{{\sin }^{2}}\frac{x}{2}=\frac{4}{5}\] Þ \[2{{\sin }^{2}}\frac{x}{2}=1-\frac{4}{5}=\frac{1}{5}\]\[\Rightarrow \sin \frac{x}{2}=\sqrt{\frac{1}{10}}\].
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