Question
Evaluate:
\[ \tan\left(\cos^{-1}\left(\frac{3}{5}\right) + \tan^{-1}\left(\frac{1}{4}\right)\right) \]
Solution
Let
\[ A = \cos^{-1}\left(\frac{3}{5}\right), \quad B = \tan^{-1}\left(\frac{1}{4}\right) \]
Find tan A:
\[ \cos A = \frac{3}{5} \Rightarrow \sin A = \frac{4}{5} \]
\[ \tan A = \frac{4}{3} \]
Find tan B:
\[ \tan B = \frac{1}{4} \]
Use identity:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 – \tan A \tan B} \]
\[ = \frac{\frac{4}{3} + \frac{1}{4}}{1 – \frac{4}{3}\cdot\frac{1}{4}} \]
\[ = \frac{\frac{16 + 3}{12}}{1 – \frac{1}{3}} = \frac{19/12}{2/3} \]
\[ = \frac{19}{12} \cdot \frac{3}{2} = \frac{19}{8} \]
Final Answer:
\[ \boxed{\frac{19}{8}} \]
Key Concept
Convert inverse trig into triangle values and apply tan(A + B) identity.