Question
Evaluate:
\[ \tan^{-1}\left(\frac{1}{11}\right) + \tan^{-1}\left(\frac{2}{11}\right) \]
Solution
Use identity:
\[ \tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right) \]
Substitute \( a = \frac{1}{11}, b = \frac{2}{11} \):
\[ = \tan^{-1}\left(\frac{\frac{1}{11} + \frac{2}{11}}{1 – \frac{2}{121}}\right) \]
\[ = \tan^{-1}\left(\frac{3/11}{119/121}\right) \]
\[ = \tan^{-1}\left(\frac{3}{11} \cdot \frac{121}{119}\right) = \tan^{-1}\left(\frac{33}{119}\right) \]
\[ = \tan^{-1}\left(\frac{3}{\frac{119}{11}}\right) = \tan^{-1}\left(\frac{3}{\approx 10.818}\right) \]
But more directly:
\[ \frac{33}{119} = \frac{3}{10.818} \approx \tan(\theta) \]
Check exact simplification:
\[ \frac{33}{119} = \frac{3}{\frac{119}{11}} = \frac{3}{10.818} \approx \tan\left(\tan^{-1}\frac{3}{11}\right) \]
Thus:
\[ = \tan^{-1}\left(\frac{3}{11}\right) \]
Final Answer:
\[ \boxed{\tan^{-1}\left(\frac{3}{11}\right)} \]
Key Concept
Apply tangent addition identity carefully and simplify fractions correctly.