Question

\( 2\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1\left(\frac{1}{4}\right) = \)}

• A. \( \tan^{-1}\left(\frac{16}{13}\right) \)
• B. \( \tan^{-1}\left(\frac{17}{23}\right) \)
• C. \( \frac{\pi}{4} \)
• D. \( 0 \)
• E. \( \tan^{-1}\left(\frac{11}{12}\right) \)

Hint:

Apply inverse tangent addition formula step-by-step for multiple angles.

Answer 1

The Correct Option is A

Concept: Use: \[ \tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right) \]

Step 1: Let \(x = \tan^{-1}(1/3)\)
Then: \[ 2x = \tan^{-1}\left(\frac{2(1/3)}{1-(1/3)^2}\right) \]

Step 2: Simplify
\[ = \tan^{-1}\left(\frac{2/3}{1-1/9}\right) = \tan^{-1}\left(\frac{2/3}{8/9}\right) = \tan^{-1}\left(\frac{3}{4}\right) \]

Step 3: Add remaining term
\[ \tan^{-1}(3/4) + \tan^{-1}(1/4) \]

Step 4: Apply formula again
\[ = \tan^{-1}\left(\frac{3/4+1/4}{1-(3/4)(1/4)}\right) \] \[ = \tan^{-1}\left(\frac{1}{1-\frac{3}{16}}\right) = \tan^{-1}\left(\frac{1}{\frac{13}{16}}\right) = \tan^{-1}\left(\frac{16}{13}\right) \]

Step 5: Final Answer
\[ \boxed{\tan^{-1}\left(\frac{16}{13}\right)} \]