| Assertion [A]: The value of \[{{\sin }^{-1}}\left( \cos \left( {{\sin }^{-1}}\frac{1}{2} \right) \right)\] is \[\frac{\pi }{3}\]. |
| Reason [R]: The value of \[{{\sin }^{-1}}\left( \cos \,x \right)\] is x. |
A) Both A and R are individually true and R is the correct explanation of A.
B) Both A and R are individually true and R is not the correct explanation of A.
C) 'A' is true but 'R' is false
D) 'A' is false but 'R' is true
E) Both A and R are false.
Correct Answer: B
Solution :
\[{{\sin }^{-1}}\left( \cos \left( {{\sin }^{-1}}\left( \frac{1}{2} \right) \right) \right)={{\sin }^{-1}}\left( \cos \left( \frac{\pi }{6} \right) \right)\] \[={{\sin }^{-1}}\left( \frac{\sqrt{3}}{2} \right)=\frac{\pi }{3}\] Reason (R) : \[{{\sin }^{-1}}\left( \cos \,x \right)={{\sin }^{-1}}\left( \sin \left( \frac{\pi }{2}-x \right) \right)\] \[=\frac{\pi }{2}-x\]. Here statement A is true but statement R is not correct explanation of statement \[\therefore \] Correct answer b.
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