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Question
The value of \[\tan\left( \cos^{- 1} \frac{3}{5} + \tan^{- 1} \frac{1}{4} \right)\]
Options
`19/8`
`8/19`
`19/12`
`3/4`
Solution
\[\tan\left( \cos^{- 1} \frac{3}{5} + \tan^{- 1} \frac{1}{4} \right) = \tan\left( \tan^{- 1} \frac{\sqrt{1 - \frac{9}{25}}}{\frac{3}{5}} + \tan^{- 1} \frac{1}{4} \right)\]
\[ = \tan\left( \tan^{- 1} \frac{\frac{4}{5}}{\frac{3}{5}} + \tan^{- 1} \frac{1}{4} \right)\]
\[ = \tan\left( \tan^{- 1} \frac{4}{3} + \tan^{- 1} \frac{1}{4} \right)\]
\[ = \tan\left( \tan^{- 1} \frac{\frac{4}{3} + \frac{1}{4}}{1 - \frac{1}{3}} \right)\]
\[ = \frac{\frac{16 + 3}{12}}{\frac{2}{3}}\]
\[ = \frac{19}{8}\]
Hence, the correct answer is option (a).
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