A) \[\frac{1}{2}{{\cos }^{-1}}\left( \frac{3}{5} \right)\]
B) \[\frac{1}{2}{{\sin }^{-1}}\left( \frac{3}{5} \right)\]
C) \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{3}{5} \right)\]
D) \[{{\tan }^{-1}}\left( \frac{1}{2} \right)\]
Correct Answer: A
Solution :
\[{{\tan }^{-1}}\frac{1}{4}+{{\tan }^{-1}}\frac{2}{9}={{\tan }^{-1}}\left( \frac{(1/4)+(2/9)}{1-(1/4)\times (2/9)} \right)\] =\[{{\tan }^{-1}}\left( \frac{1}{2} \right)=\frac{1}{2}.2{{\tan }^{-1}}\left( \frac{1}{2} \right)=\frac{1}{2}{{\tan }^{-1}}\frac{2(1/2)}{1-(1/4)}\] \[=\frac{1}{2}{{\tan }^{-1}}\frac{4}{3}=\frac{1}{2}{{\cos }^{-1}}\frac{3}{5}\].
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