| Assertion [A]: The value of \[{{\tan }^{-1}}\sqrt{3}-{{\sec }^{-1}}(-2)\] is \[\frac{\pi }{3}\]. |
| Reason [R]: If \[{{\cos }^{-1}}x=y\] then principal value of y is \[0\,\le \,\,y\,\,\le \,\,\pi \]. |
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: D
Solution :
\[{{\tan }^{-1}}\sqrt{3}-{{\sec }^{-1}}\left( -2 \right)\] \[={{\tan }^{-1}}\left( \tan \frac{\pi }{3} \right)-\left\{ \pi -{{\sec }^{-1}}\,2 \right\}\] \[=\frac{\pi }{3}-\pi +{{\sec }^{-1}}\left( \sec \frac{\pi }{3} \right)=-\frac{2\pi }{3}+\frac{\pi }{3}=-\frac{\pi }{3}\] Here Assertion A is false and Reason R is true. \[\therefore \]Option [D] is the correct answer.
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