A) \[\frac{17}{25}\]
B) \[-\frac{51}{135}\]
C) \[\frac{-2\sqrt{18}}{120}\]
D) \[\frac{9\sqrt{21}}{125}\]
Correct Answer: D
Solution :
Given, \[\cos \,\left( 3{{\sin }^{-1}}\left( \frac{2}{5} \right) \right)\] Let \[{{\sin }^{-1}}\,\left( \frac{2}{5} \right)=\theta \,\,\Rightarrow \,\,\sin \theta =\frac{2}{5}\] \[\Rightarrow \] \[\cos \theta =\frac{\sqrt{21}}{5}\Rightarrow \theta ={{\cos }^{-1}}\left( \frac{\sqrt{21}}{5} \right)\] \[\Rightarrow \] \[\cos \,\left( 3\,{{\cos }^{-1}}\left( \frac{\sqrt{21}}{5} \right) \right)\] \[=\cos \,\left[ {{\cos }^{-1}}\left( 4\times {{\left( \frac{\sqrt{21}}{5} \right)}^{3}}-3\times \frac{\sqrt{21}}{5} \right) \right]\] \[\left[ \because \,\,3\,\,{{\cos }^{-1}}x={{\cos }^{-1}}(4{{x}^{3}}-3x),\,x\in \left[ \frac{1}{2},1 \right] \right]\] \[=4\times \frac{21\sqrt{21}}{125}-\frac{3\sqrt{21}}{5}=\frac{84\sqrt{21}-75\sqrt{21}}{125}=\frac{9\sqrt{21}}{125}\]You need to login to perform this action.
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