| Assertion [A]: The value of \[\sin \left( {{\tan }^{-1}}\frac{a}{b} \right)\] is \[\frac{a}{\sqrt{{{b}^{2}}-{{a}^{2}}}}\]. |
| Reason [R]: The value of \[\tan \left( {{\cot }^{-1}}\frac{a}{b} \right)\] is \[\frac{b}{a}\]. |
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 :
\[\sin \left( {{\tan }^{-1}}\frac{a}{b} \right)=\sin \left( {{\sin }^{-1}}\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)=\frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\] Reason (R) : \[\tan \left( {{\cot }^{-1}}\frac{a}{b} \right)=\tan \left( {{\tan }^{-1}}\frac{b}{a} \right)=\frac{b}{a}\]. Here Assertion [A] is false and Reason [R] is true. \[\therefore \]Option [D] is the correct answer.
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