Correct Answer - Option 2 :
\(\frac{{2\sqrt 6 }}{5}\)
Concept:
\({\cos ^{ - 1}}\left( {\cos x} \right) = x\)
Calculation:
\(\cos \left( {{{\sin }^{ - 1}}\frac{1}{5}} \right)\)
Let,
\({\sin ^{ - 1}}\frac{1}{5} = x\)
\(\sin x = \frac{1}{5}\)
We know,
\({\sin ^2}x + {\cos ^2}x = 1\)
\({\left( {\frac{1}{5}} \right)^2} + {\cos ^2}x = 1\)
\({\cos ^2}x = 1 - {\left( {\frac{1}{5}} \right)^2}\)
\({\cos ^2}x = \frac{{24}}{{25}}\)
\(\cos x = \frac{{2\sqrt 6 }}{5}\)
\(x = {\cos ^{ - 1}}\frac{{2\sqrt 6 }}{5}\)
Now substitute for \({\sin ^{ - 1}}\frac{1}{5}\) in \(\cos \left( {{{\sin }^{ - 1}}\frac{1}{5}} \right)\)
\(\cos \left( {{{\sin }^{ - 1}}\frac{1}{5}} \right) = \cos \left( {{{\cos }^{ - 1}}\frac{{2\sqrt 6 }}{5}} \right)\)
\( = \frac{{2\sqrt 6 }}{5}\)
