Question
Evaluate:
\[ \tan\left(2\tan^{-1}\left(\frac{1}{5}\right)\right) \]
Solution
Let
\[ \theta = \tan^{-1}\left(\frac{1}{5}\right) \Rightarrow \tan \theta = \frac{1}{5} \]
Using identity:
\[ \tan 2\theta = \frac{2\tan \theta}{1 – \tan^2 \theta} \]
Substitute:
\[ \tan 2\theta = \frac{2 \cdot \frac{1}{5}}{1 – \left(\frac{1}{5}\right)^2} \]
\[ = \frac{2/5}{1 – 1/25} = \frac{2/5}{24/25} \]
\[ = \frac{2}{5} \cdot \frac{25}{24} = \frac{50}{120} = \frac{5}{12} \]
Final Answer:
\[ \boxed{\frac{5}{12}} \]
Key Concept
Use the identity \( \tan 2\theta = \frac{2\tan\theta}{1 – \tan^2\theta} \) to simplify expressions.