Question

\( \tan^{-1} \left( \frac{1}{4} \right) + \tan^{-1} \left( \frac{1}{9} \right) \) equals to:

• A. \( \frac{1}{2} \cos^{-1} \left( \frac{3}{5} \right) \)
• B. \( \frac{1}{2} \sin^{-1} \left( \frac{3}{5} \right) \)
• C. \( \frac{1}{2} \tan^{-1} \left( \frac{3}{5} \right) \)
• D. \( \tan^{-1} \left( \frac{1}{5} \right) \)

Hint:

Use the formula for the sum of inverse tangents to simplify expressions involving \( \tan^{-1} \).

Answer 1

The Correct Option is D

Step 1: Use the formula for the sum of inverse tangents.
We use the formula for the sum of inverse tangents: \[ \tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x + y}{1 - xy} \right) \] Substituting \( \frac{1}{4} \) and \( \frac{1}{9} \), we get the answer as \( \tan^{-1} \left( \frac{1}{5} \right) \).
Thus, the correct answer is (4).