A) \[\frac{13}{6}\]
B) \[\frac{17}{6}\]
C) \[-\frac{13}{6}\]
D) \[-\frac{17}{6}\]
Correct Answer: B
Solution :
\[\tan \left[ {{\cos }^{-1}}\frac{4}{5}+{{\tan }^{-1}}\frac{2}{3} \right]\] \[=\tan \left[ {{\tan }^{-1}}\frac{\sqrt{1-\frac{16}{25}}}{\frac{4}{5}}+{{\tan }^{-1}}\frac{2}{3} \right]\] \[\left[ \because \,{{\cos }^{-1}}x={{\tan }^{-1}}\frac{\sqrt{1-{{x}^{2}}}}{x} \right]\] \[=\tan \left[ {{\tan }^{-1}}\frac{3}{4}+{{\tan }^{-1}}\frac{2}{3} \right]\] \[=\tan \left[ {{\tan }^{-1}}\left( \frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}.\frac{2}{3}} \right) \right]\] \[=\tan \left[ {{\tan }^{-1}}\left( \frac{17}{6} \right) \right]\] \[=\frac{17}{6}\]
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