| Assertion [A]: The principal value of \[{{\cos }^{-1}}\left( \cos \frac{13\pi }{6} \right)\]is \[\frac{\pi }{6}\]. |
| Reason [R]: The principal value of \[{{\tan }^{-1}}\left( \tan \,\frac{5\pi }{6} \right)\] is \[\frac{5\pi }{6}\]. |
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: C
Solution :
\[{{\cos }^{-1}}\left( \cos \frac{13\pi }{6} \right)={{\cos }^{-1}}\left( \cos \left( 2\pi +\frac{\pi }{6} \right) \right)\] \[={{\cos }^{-1}}\left( \cos \frac{\pi }{6} \right)=\frac{\pi }{6}\]. Reason (R) : \[{{\tan }^{-1}}\left( \tan \frac{5\pi }{6} \right)={{\tan }^{-1}}\left( \tan \left( \pi -\frac{\pi }{6} \right) \right)\] \[={{\tan }^{-1}}\left( -\tan \frac{\pi }{6} \right)=-\frac{\pi }{6}\]. Here Assertion [A] is true and Reason [R] is false. \[\therefore \]Option [C] is the correct answer.
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