question_answer

Value of \[\cos \,\left( {{\sin }^{-1}}\left( \frac{2}{5} \right) \right)\] is

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 \[{{\sin }^{-1}}\,\left( \frac{2}{5} \right)=\theta \,\,\Rightarrow \,\,\sin \theta =\frac{2}{5}\] \[\Rightarrow \] \[\cos \theta =\frac{\sqrt{21}}{5}\Rightarrow \theta ={{\cos }^{-1}}\left( \frac{\sqrt{21}}{5} \right)\] \[\Rightarrow \] \[\cos \,\left( 3\,{{\cos }^{-1}}\left( \frac{\sqrt{21}}{5} \right) \right)\] \[=\cos \,\left[ {{\cos }^{-1}}\left( 4\times {{\left( \frac{\sqrt{21}}{5} \right)}^{3}}-3\times \frac{\sqrt{21}}{5} \right) \right]\] \[\left[ \because \,\,3\,\,{{\cos }^{-1}}x={{\cos }^{-1}}(4{{x}^{3}}-3x),\,x\in \left[ \frac{1}{2},1 \right] \right]\] \[=4\times \frac{21\sqrt{21}}{125}-\frac{3\sqrt{21}}{5}=\frac{84\sqrt{21}-75\sqrt{21}}{125}=\frac{9\sqrt{21}}{125}\]