Question

The value of $\cos \left( {{{\sin }^{ - 1}}\left( {\dfrac{2}{3}} \right)} \right)$ is equal to:
A. $\dfrac{{\sqrt 3 }}{5}$
B. $\dfrac{5}{3}$
C. $\dfrac{5}{{\sqrt 3 }}$
D. $\sqrt {\dfrac{5}{3}} $
E. $\dfrac{{\sqrt 5 }}{3}$

Answer 1

Hint: Use the information, ${\sin ^{ - 1}}x = {\cos ^{ - 1}}\sqrt {1 - {x^2}} $. Convert in same, inverse-trigonometric ratios.

Complete step-by-step answer:

We have given that, $\cos \left( {{{\sin }^{ - 1}}\left( {\dfrac{2}{3}} \right)} \right)$. We know that, ${\sin ^{ - 1}}x = {\cos ^{ - 1}}\sqrt {1 - {x^2}} $. Using this identity, we get, $\cos \left( {{{\sin }^{ - 1}}\left( {\dfrac{2}{3}} \right)} \right) \Rightarrow \cos \left( {{{\cos }^{ - 1}}\sqrt {1 - {{\left( {\dfrac{2}{3}} \right)}^2}} } \right)$.
On further solving it we get,
\[
  \cos \left( {{{\sin }^{ - 1}}\left( {\dfrac{2}{3}} \right)} \right) \\
   \Rightarrow \cos \left( {{{\cos }^{ - 1}}\sqrt {1 - {{\left( {\dfrac{2}{3}} \right)}^2}} } \right) \\
   \Rightarrow \cos \left( {{{\cos }^{ - 1}}\sqrt {1 - \dfrac{4}{9}} } \right) \\
   \Rightarrow \cos \left( {{{\cos }^{ - 1}}\sqrt {\dfrac{5}{9}} } \right) \\
   \Rightarrow \sqrt {\dfrac{5}{9}} \\
   \Rightarrow \dfrac{{\sqrt 5 }}{3} \\
\]
Hence the correct option is E.

Note: In inverse trigonometric problems, identities and formulas are the key. Our first approach should be, which formula we have to apply in order to simplify the equation. When you get the correct formula, half of the question is finished there itself.