A) \[\frac{\sqrt{5}}{3}\]
B) \[\frac{2\sqrt{5}}{3}\]
C) \[\frac{4\sqrt{5}}{9}\]
D) \[\frac{2\sqrt{5}}{9}\]
Correct Answer: C
Solution :
We have, \[\sin \left[ 2{{\cos }^{-1}}\frac{\sqrt{5}}{3} \right]\] \[=\sin \left[ {{\cos }^{-1}}\left( 2\cdot {{\left( \frac{\sqrt{5}}{3} \right)}^{2}}-1 \right) \right]\] \[[\because \,\,2{{\cos }^{-1}}x={{\cos }^{-1}}(2{{x}^{2}}-1)]\] \[=\sin \left[ {{\cos }^{-1}}\left( \frac{1}{9} \right) \right]\] \[=\sin \left[ {{\sin }^{-1}}\sqrt{1-{{\left( \frac{1}{9} \right)}^{2}}} \right]\] \[[\because \,\,{{\cos }^{-1}}x={{\sin }^{-1}}(\sqrt{1-{{x}^{2}}})]\] \[=\sin \left[ {{\sin }^{-1}}\sqrt{\frac{80}{81}} \right]\] \[=\frac{\sqrt{80}}{9}\] \[=\frac{4\sqrt{5}}{9}\]
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